P UBLICATIONS DU D ÉPARTEMENT DE MATHÉMATIQUES DE LYON
M ANABU H ARADA
Applications of Factor Categories to Completely
Indecomposable Modules
Publications du Département de Mathématiques de Lyon, 1974, tome 11, fascicule 2
p. 19-104
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APPLICATIONS OF FACTOR CATEGORIES TO COMPLETELY
INDECOMPOSABLE MODULES
by Manabu HARADA
In this note we assume the reader is familiar to elementary properties of rings and modules. In some sense we can understand that the
theory of categories is a generalization of the theory of rings.
Especially, additive categories A have very similar properties to rings
from their definitions.
From this point of view, we shall define an ideal C in A and a
factor category A/£
of A with respect to C (see Chapter 1 ) , which is
analogous to factor modules or rings. The purpose of this lecture is to
apply those factor categories to completely indecomposable modules.
First, we take an artinian ring R. The radical J(R) of R is a very
important tool to study structures of R. Since R/J(R) is a semi-simple
and artinian ring, we know useful properties of R/J(R). In order to
study structures of R, we contrive to lift those properties to R. The
idea in this note is closely related to the above situation.
19
Applications of Factor Categories . . .
Let R be a ring with identity and (M }^ a set of completely inde­
composable right R-modules. In Chapter 1 we define the induced category
A from {M^} , which is a full sub-additive category in the category
1
of all right R-modules and define a special ideal JJ of A. Then A/J is
a abelian Grothendieck and completely reducible category (Theorem 1 . H . 8 ) ,
which is nearly equal to Mg, where S is a semi-simple artinian ring. In
this note we frequently make use of this theorem. Especially, in Chapter
2 we shall prove the Krull-Remak-Schmidt-Azumaya theorem by virtue of
1
this theorem, (see below).
Let { M } and {N }
fi
such that M =
T
be any sets of completely indecomposable modules
I ® M = I ® N . Then we consider the following properties :
a
j
B
Q
T
I) There exists a one-to-one map-ping <j> of I to J such that M v N . , ,
a
<Ma)
and hencej \l\ = |j| where |l| means the cardinal of I.
3
!
II) (Take out (some components)) For any subset I of I, there
1
exists a one-to-one mapping
a' eV
and M =
M
N
of I into J such that »^ ^( »)
a
a
f
or
ZZ
.* N , , $
£1 « M „ .
a'€i»
a €l-I'
}
f
a
f
1
II ) (Put into) For any subset I of I, there exists a one-to-one
1
mapping
M =
C
a £I
f
of I* into J s^c/? t/zat M <fcN^ ,j for a'si and
al
• M
f
a
•
JZ
3^J-^(I')
$ NQ, .
p
20
a
Applications of Factor Categories . . .
Ill) Every direct summand of M
is also a direct sum of completely
indecomposable modules.
1
M has always the properties I), II) and II ) if I
!
1
in II) and II )
1
are finite, which we call the Krull-Remak-Schmidt-Azumaya theorem. If
1
1
it is allowed to take any subset I , in II) or II )* then it is clear
that II) and II') are equal to each other.
1
G. Azumaya [i\ proved the avove II) and II ) step by step and
f
proved I) with II) and I I )
5
provided I
1
is finite. We shall prove them
independently and its proof suggests us how we can drop the assumption
of finiteness on I
1
f
in the Azumaya theorem. This argument is very much
owing to the factor category A/J
1
. The idea of dropping the assumption
of finiteness gives us a definition of locally semi-T-nilpotency of the
set of
(M )j (see Chapter 2 ) , which is a generalization of T-nilpotency
defined by H. Bass [2] .
On the other hand, the exchange property is very important to
study decompositions of modules (cf. [Ul). In this note we shall slightly
change its definition as follows : Let M be an R-module and N a direct
summand of M. We suppose that for any decomposition M =
\I\ 4
E 9 Kg with
a, we have a new decomposition ; M = N $ E $ Kg , where Kg SKg
for all 8 el. In this case, we say N has the ^.-exchange property in M.
If N has the a^xch&nge property in M for any cardinal a, we say N has
the exchange property in M. Furthermore, we define a new concept in
21
Applications of Factor Categories . . .
Chapter 3 . Let K be a submodule of M and K = I © K , . If for any
J'
finite subset J' of J Z ® K , is a direct summand of M, we call K
J
a locally direct summand of M (with respect to the decomposition
Y
1
Y
K = Z © K )• It is clear that if all K are injective, K is always a
J
locally direct summand of M. This property is useful to consider the
Y
Y
problem of Matlis
f 29]
, which is the property III) in case of inject
modules.
Those concepts are mutually related in the following theorem
3 . 1 . 2 and 3 . 2 . 5 )
(Theorems
: Let M and {M } be as above. Then the
06 J-
following statements are equivalent.
1) M satisfies the take out property of any subset V of I and
for any (Nglj .
2) Every direct summand of M has the exchange property in M.
3) ^ ^ j ^
M
s a
a
locally semi-T-nilpotent system.
h) Every locally direct summand of M is a direct summand of M.
5) J n End^M) is equal to the Jacobson radical J of EndpCM).
1
6) Endp(M)/J
is a regular ring in the sense of Von Neumann and
every idempotents in End^(M)/J are lifted to End^(M).
22
Applications of Factor Categories . . .
We study the propery III in Chapters 3 and k and give a special answer
for it, even though it is not complete, (Theorem 3 . 2 . 7 ) ,
(cf.[6,7,17,
18,2^,381).
In 1960 H. Bass [2]
defined (semi-) perfect rings as a generalization
of semi-primary rings and E. Mares [28] further generalized them to
(semi-} perfect modules in 1963• In Chapter 5 we shall prove the following
theorem (Theorem 5 . 2 . 1 )
P =
Z $ P
I
; let {^^j be
a
s e t
°f projective modules and
. Then J(P) is small in P if and only if J(P ) is small
01
a
in P^ for all a d
and
s
{P^lj ^
& locally semi-H-nilpotent system.
Using this theorem and Mares' results, we shall study structures of
(semi-) perfect modules.
In Chapter 6 we shall study injective modules. Let {E }- be a set
of injective modules and B the induced category from {E^} . First we
shall prove that B/J
is an abelian Grothendieck and spectral categoryj
where J is the radical of B
(Theorem 6 . 2 . 1 ) . We shall study decompo­
sitions of injective modules by making use of this theorem
(cf.
[ 1 0 , 2 9 , 3 l ] ) . Finally we shall consider the Matlis'problem (cf. [ 9 , 1 2 ,
2 5 , 38,U0,Ul] ). Relating to it, we shall give the following theorem
(Theorem 6 . 5 . 3 )
; Let (E
modules, E = I $ E
I
and A
be a set of injective and indecomposable
1
the induced category from the all completely
01
indecomposable modules. Then the following statements are equivalent.
1) {E }-
23
is a locally semi-T-nilpotent
system.
Applications of Factor Categories . . .
2) Every module M in A
!
which is an extension of E
1
which is an essential extension of E
contains E
as a direct surnmand.
3) Every module M in A
coincides with E.
k) For any monomorphism f in End^ÍE) Im f is a direct surnmand
of E.
This lecture note gives some applications of the theory of category
to the theory of modules, however conversely we can apply some concepts
in this note to special categories and define
semi-perfect or semi-
artinian Grothendieck categories, which preserve many properties of
semi-perfect or semi-artinian rings
(see [ 2 2 ! ) .
This lecture was given at Universidad national del Sul in Argentina
and The University of Leeds in England and the first part was given at
Universite Claude Bernard Lyon-1 in France in 1 9 7 3 . The author would
like to express his heartful thanks to those universities for their kind
invitations and hospitalites and to Université de Lyon for publication
of this note.
24
Applications of F actor Categories
CHAPTER 1 . A PRINCIPAL THEOREM
We shall assume the reader has some knowledge about
elementary
definitions and properties of modules and categories. We refer to
f1 1 , 3 c j
for them.
1 . 1 . IDEALS.
We always study
additive categories A and so we shall assume that
categories in this note are additive, unless otherwise stated. We shall
use the following notations :
; the category of all right R-modules, where R is a ring with
identity.
A
; the class of all morphisms in A.
For a, 8 in A^ " aB is defined" implies codomain of 8 = domain of a
and
"a±8 is defined" implies domain of a = domain of 8 and codomain of
a = codomain of 8.
We shall define ideals in an additive category A.
DEFINITION.- Let C be a subclass of A . If C satisfies the following
—
—m
—
conditions, C is called a left ideal of A.
1 . For any oteA and 8 e C
—m
—
if aB is defined, a B e C
—
2 . For any y^6 G £, if Y±6
is defined, y ± 6 € C , (cf. [ 5 ] ) .
We can define similarly right or two-sided ideals in A. Let £ be
a two-sided ideal in A. If [A,A] f\ C is the Jaccbson radical of fA,A] for
all AC A, C iscalled the Jaaobson radical of A, (if A has finite co-products,
25
Applications of Factor Categories . . .
the Jacobson radical is uniquely determined, (see [ l 6 , 2 T ] ) ) .
The following notion is essential in this note.
DEFINITION.^Let A he an additive category and £ a two-sided ideal in A.
We define a factor category A/£ of A with respect to C as follows :
1
The objects in A/£ coincide with those in A (for A in A , A means
that A is considered in A/C).
For A,B€A/£,
2
residue class of f in
(for f £ fA,B] ,f means the
[A,BJ = [ A , B ] / [ A , B ] O C
[A,BJ /[A,BJ
O C) .
Remarks 1.-It is clear A/C is also an additive category. In general even
if A is abelian, A/£ is not abelian. If we want to use structures of
factor categories, we should find good ideals £ such that A/£ become good
categories.
2.
n
Let A = ¡2 $ A. in A. Then there exists inclusions i, and
i=1
projections p
k
such that 1
1
A
K
=
P \ = ^
k
and i^p = 0 if j^k.
k
relations are preserved in A / £ , i.e. 1 ^ = £
Those
=
» ^k^k ^A^
and i.p = 0 if j^k. Hence, A = L $ A. in A/£. This is not true for infij
K
1
nite coproducts.
3 . If A,B are isomorphic each other in A, then there exist morphisms a : A
B and 6 : B
>A such that
aB = 1 - and 8a = 1 .
B
A
A
Hence, A,B are isomorphic each other in A/£. However the converse is not
true, in general. If £ is the Jacobson radical, the converse is also true.
Because, if A,B are isomorphic, there exist a : A — ? B, 8
f
26
!
:B
*A
Applications of Factor Categories . . .
f
1
such that a*B = 1^ and B'a' = 1 . Hence, l.-B'a
A
n
TA,A] . Therefore,
[B,B]. Hence, a J 6
?
PROPOSITION 1 . 1 . 1 .
A
f
g a
!
is in the radical of
A
f
is a unit in [A,A] • Similarly, a B
f
is a unit in
are isomorphisms.
- Let A,B
be additive categories and T : A — * B
*)
an additive covariant functor. Then C
—
sided ideal in A and T = T i^
0
y
= {a|€ A , Ta=0}
is a two-
""El
where, ij; : A —* A/£ ts a natural
functor and T : A/£ — * B is
naturally induced from T.
1 . 2 . ABELIAN CATEGORIES.
Let A he an additive category. There are many equivalent definitions
for A to be
abelian. We shall take the following :
i
For any two objects A,B in A the coproduct A © B of A and
B is defined and belongs to A.
ii A contains a zero object (so does an additive category).
iii
For each morphism f in A Ker f and Coker f exist in A.
iv
(normal) For each monomorphism f in A, f is a kernel of
;
some morphism in A.
iv
1
(concrmal) For each epimorphism f in A, f is a cokernel
of some morphism in A.
In this section, we shall rewrite the above definition of an abelian
category by virtue of another terminologies, which are very familiar to
the ring theory.
*) In general, it is not a set, but we shall use the same notation as the
set. We always use such notations.
27
Applications of Factor Categories.. .
Let A be an additive category and S a subclass of A . We put
—
~-m
(6|e A , a6 is defined and a6 ^ S}
—m
(S:a) =
r
, (S:a) = {B|cA ,
i
—m
8a is defined and 8a € S}. If (0:a) ^0 for some a c A , a is called a left
r
—m
zero-divisor. Similarly, we define
a right zero-divisor. From the
definitions, we know that a is monomorphic (epimorphic) if only only if
a'
a
a is not left (ri^ht) zero-divisor. Let C
y A
> B t>e a sequence.
Then a
is the kernel of a if and only if ( 0 : a ) =0 and (0;a) =a A ,
r
—m
where a'A^ = { a y|yq A^ , a y is defined}, a is the cokernel of a if and
!
1
f
r
r
only if ( 0 : a )
=0 and
(0;a )-=A a .
!
i —m
1
1 . 2 . 1 . - L e t A be an additive category with, finite co-products.
PROPOSITION
Then A is abelian if and only if A satisfies the following conditions
1 For each a * A , there exists 8 ^ A such that ( 0 : 8 ) =0 and
—m
-in
r
(0:a) =8A .
r
—m
2 For each a s A
—m
there exists 8
!
f
s
such that ( 0 : B ) - 0 and (0:aL
I
1
3 For each ye A such that (0:y) = 0 , (0:(0:y) J = yA .
-m
r
1 r
-m
[
4 For each y'£A
Proof. -
such that (0:y ) = 0 , (0:(0:y») ) =A y .
f
f
By the assumption A^ satisfies i, ii in the above definition
and iii corresponds to 1 , 2 from the above remark. We assume A is abelian
Let y
be as in 3 . Then there exists a cokernel 8 of y ; 0—*A — •••-*B — ^ C - * 0
exact. Then y = KerB and B = Coker y . Henc*?, ( 0 : 6 )
and (0:y) =A 8
r
m
1 —m
from the above remark. Therefore, ( 0 : ( 0 : y ) - ) =(0:A 8) = ( 0 : 8 ) =YA .
l r
—m r
r —m
= yA
M
28
Applications of Factor Categories . . .
k is dual to 3 . Conversely, we assume A satisfies 1 ~ h. We know from
—m
the remark that 1 , 2 guarantee the existence of kernel and cokernel for
any ot£ A^. Let Y'-A—>B be monomorphic . Then there exists 8 A^ such that
€
6
is epimorphic and (0:y) -|
=
^ 8 from 2 . Furthermore, ( 0 : ( 0 : y ) )
=
1
r
(0:A 6) = ( 0 : 8 ) = Y A by 3 . Hence, y = Ker 8 and we have iv. iv is
-m r
r
-m
f
dual to iv. Therefore, A is abelian,
1 . 3 . AMENABLE CATEGORIES.
We shall define some special categories which we shall use later.
DEFINITION. Let A be an additive category. A is called regular if [A,Al
is a regular ring in the sense of Von Neumann for all A€ A. A is called
amenable if A has finite co-products and for any idem potent e in
LA,AJ splits, i.e. A = Im e 9 Ker e for all AG A, (see [ 1 1 ) ). A
c
called
spectral if all f ^
PROPOSITION 1 . 3 . 1 . - Let A
Then A
m
is
splits (see f l 3 ] ) .
be an additive amenable and regular category.
3
is abelian.
Proof. - Since A is amenable, A satisfies the assumption in ( 1 . 2 . 1 ) . We
shall show A satisfies 1
Put a
f
in ( 1 . 2 . 1 ) . Let a:A-*B be monomorphic.
= (°
°| : A©B
ASB. Since A is regular, there exists
a
x = (x^j) ^ [A® > ^ ] such that a xa = a'. Hence, a =
•
V
]
0
B
A
B
f
?
a x
a
P u t
1 2
2
e = x a , then e=e
10
and ae=a. Hence, A a = A
29
e. Since A is amenable,
Applications of Factor Categories...
1
e = i^e
, where e
f
: A—^Im e is epimorphic and i^En e -> A is the
inclusion. Thus, we have (0:a) = (0:A a) * (0:A e) = (0:e) - (l -e)A C
r
in r
—m r
r
A
—m —
A
i( _ )A ^(0:a) .Hence(0:a) =i _ A and ( 0 : (0:o) ) ,-(0: i , _ )
1
A
n
e
r
r
(l
e)
m
r
(
=
e )
a, which gives 2 and h in ( 1 . 2 . 1 ) . From the duality we obtain 1 and 3 .
Therefore, A is abelian.
We can easily see from the above proof that Im e = Im a. Thus, we
have
COROLLARY 1 . 3 . 2 f35] . Let A be an additive and amenable category, Then
A
is (abelian) spectral if and only if A is (abelian) regular.
1.U A principal theorem on indecomposable modules
Let R be
a ring with identity. We consider always unitary right
R-modules M. If End^(M) is a local ring (i.e. its radical is a unique max
maximal left or right ideal), M is called completely indecomposable module
(briefly c.inde.). It is clear that c.inde. module is indecomposable as
a directsum, however the converse is not true. We note that the radical
is equal to the set of all non-isomorphisms in End^(M) if M is c.inde.
by the following.
LEMMA. I.U.I. - Let M. , i = 1 . 2 . 3 be (c.) inde. and f.:M.
I
'
1
•
R-homomorphisms for i = 1 , 2 . if
Proof. - Since fgf^ is isomorphic,
Furthermore, Mg = Im
f^f
1
M. ,
i+1
is isomorphic, f^ are isomorphic.
is monomorphic and fg is epimorphic.
« Ker fg. Hence, Ker f
30
2
= 0 and Im
= M.
g
Applications of Factor Categories...
Let {M }
Ot
T
, {N }
0
1
p
be sets of modules and put M = £ « M
T
d
^
and
N = I ® N .
Q
j
Ct
P
We shall describe Hom^MjN) as the set of matrices. Let a.. : M. —* N.
n
ij
j
i
be R-homomorphisms. If I and J are finite, Hom^(M,N) = {(J* I) matrices (ot^ ^)}.
We assume I and J are infinite. Let m be an element in
and fc Hom^(M,N).
n
Then f(m) = Y
n . ; n .£N .. From this remark, we can define
0
0
pi
6
fl
I
$i
*
a summable set of homomorphisms {a.-j}. as follows : for any m in
J
J
a^(m) = 0 for almost all j c J. In this case Z, Oj. ^ has a meaning and
J
a - :M
N is an R-homomorphism. A matrix (a. .) is called column
j
1
IJ
i
summable if {a..}, is summable for all i c l . Then it is clear that
Ji
J
Hom^(M,N) is isomorphic to the modules of all column summable matrices
with entries a...
Let T = £ « T be another module and f sHom^M,!*), g € Hom^N.T).
K
We assume f = (a..) and g = (6 ) as above. Then we can easily show that
1J
PQ.
5
a
gf = ^pq^ ^ ij^'
T h u s
i f
>
M=N=T, End(M) is isomorphic to the ring of
all column summable matrices (a..).
Now, we shall assume that all M
,N
0
and T
are c.inde. . We
define a subset.
jt(B»a)
^
(ji(3,a,
ma
y
=
^
a
^
I
Hom_(M,N) and no one of a..
£
ij
k
depend
decompositions M and N ) .
o n
LEMMA 1.U.2. - Let M = £ 0 M
N
a
a ^
T
p
Hom (M,N) C j' P>
a)
^
and T = Z ® T
G
J
T
(
R
,N = £ 8 N
a
I
M
is isomorphic),
ij
Hom (N,T)J»
R
.
31
K
(cf5a)
and all
P
Q jt(B.a) ^ ( p , a ) .
Jt
Applicationsc of Factor C a t e g o r i e s . . .
Proof. -
Let f • (a^) c J
where x.
=
, ( a
'
a )
, h = (b. ) C Hbm (N,T) and hf = U
R
fe
t
g
) .
3
£ ^v !, • If M ^ T. , x
is not isomorphic. We suppose
XS
uK K S
S
Xi
ts
K
M ^ T\ . Let m^O be in M , Since (a. .) is column summable, there exists a
s
t
s
1J
finite subset J of J such that a. (m) = 0 if ke-J-J . Put x. = E b
a
~ks
ts
tk.^k.
j*.
f
1
K 6 J t
b
+ ^
a
tk ks* ^
e
n
either the latter nor former term is isomorphic by
the definition of J' and
Thus x
is not isomorphic by the remark
T/ s
. Hence, hf U ' '
before ( 1 . U . 1 ) and the fact M
s
P , 0 t
' , Similarly we
Xi
have the last part.
0
PROPOSITION.-1.U.3
fll The above module j^ *^
does not depend on
decompositions of M and N. Especially> if M=N, J
f
is a two-sided
ideal in Endp(M).
Proof. - Let M = £ «
and N = £ $ N
Q
= £
GN^ , . Put T = N = £
in ( 1 . U . 2 ) . Then for any f C J ' ^ ^ ' . f = 1 fej'' ' '. Therefore,
0
T f
(a,ot)
c
T l
(a* ,a)
c
.
.
.
T l
(a' ,a) -
!
J
£J
. Similarly, ve obtain J'
JI(a ,a)
(a,a)
01
T l
(a,a)
9J
?
,,
and hence
1
=
Jt
From ( 1 . U . 3 ) we denote J ^ ° ' ^ by J'.
a
We shall give here elementary properties of a ring.
LEMMA l.k.k. - Let R be a ring and e,f idempotents such that eR<ofR and
(1-e)R
(l-f)R. Then there exists a regular element a in R sucft
1
tfczt f = a ea.
32
Applications of Factor Categories.
..
Proof. - R = eR«(l-e)R = fR«(1-f)R. Therefore, 4
a
e
say <f> = a^. Then it is clear that ^ -j
=
f^,^
= (J). + <j>£ End (R)=R ,
2
R
1
means the set of the
left multiplications of elements in R ) .
We shall later make use of the following.
COROLLARY 1.U.5• - Let P be a vector space over a division ring Aj say
P = E ®u A . Let S = End^ (P) and e an idempotent in S. Then there
a
exist a subset J of I and a regular element a in S such that for
the projection f:P->Z®v A
J
Proof. - Let eP = l
e = a Va.
Y
u
I v A and we may assume P = 2 $ u A ® ^ ^ r ^
J
J
I-J
Y
Since eS
'
P
Hom^(P,eP)^ Hom^(P,fP), we 'have the corollary by (1.U.U).
Now, we shall enter into a main part of this section. Let
^ ^i
a
be a set of c.inde. Modules. By A(A^) we shall denote the full sub-additive
category in
sums V?$T
K
Y
, whose objects consist of all kinds of (finite) direct
such that T ' s are isomorphic to some M
Y
6
n
A (A^) the (finitely) induced category from tM^}^,
in {M }_. We call
a I
(we shall use the
same terminology even if (M^} are not c.inde.).
DEFINITION .-Let B be an additive category. If B satisfies the follo­
wing properties, B is called a Grothendieck category.
1 B is abelian.
2 B has any co-products.
3 Let B G B and {B^}, C sub-objects of B such that {B^} is a directed
set. Then
33
Applications of Factor Categories . . .
(UB )NC =
A
U(B
a n
c).
(This corresponds to a fact that functor Lim is exact (see f30J, Ch. 3 ) ) .
h B has a generator, (this implies B is complete (see
)).
Definition. Let B he as above. If every object in B is artinian (noetherian)
vith respect to sub-objects, B is called
artinian (noetherian). If every
object in B is a co-product of minimal objects, B is called completely
reducible. If the Jacobson radical of B is zero, B is called semi-simple.
LEMMA 1.U.6, - Let A be a semi-simple category with finite co-products.
If a.±0£
[M,N],
there exist $,
6'g[N,M]
/0
such that QatO and aB'^O.
0\
Proof. - Put P = M M , S = fp,p]and a* = |
. If
[N,M]cx =
0 , Set* is
nilpotent, which is a contradiction. Similarly, we have otLN,M| # 0 .
COROLLARY 1 . U . 7 . - Let A be as above. If [ M M ] is a division ring, M is
;
a minimal object.
Proof. -
Let M £
N.
Then
= 0 . Hence,
[M,N]
[N,M]=
0 by ( 1 . U . 6 ) and
so the inclusion map : K —^ M is zero.
From now on, by [M,N]
THEOREM 1.U.8
we shall denote Honip(M,N) for R-modules M,N.
(Principal theorem)
fl?] —Let {M ^ be a set of
c
inde. modules and A,A the induced category and finitely induced
f
category, respectively. Let J
(1.4.2). Then A/±\
1
!
be the ideal in A defined before
(A^/J ) is a Grothendieck and completely reducible
(completely) category.
34
Applications of Factor Categories . . .
1
Proof. - We put A = A/J' (A^ = A^/J ). From the definition of co-product
and ( 1 . U . 3 ) we can easily show
I « M
= £ $ M . Put S
Y
= [M,M]/[M,MJOJ
J _
J
in A. Then S = {(& )» column finite }, since
for an object M = £
M
aT
a
°
= 0 for almost all o. We rearrange M as follows : M = £ Z © M „ ;
a I »B
[Z
•M
a
A
G
, E • M gl - {(x ) |
a
f
Y
a
ag
(\
column finite and
.M^
P
= A ,
a
a
which is a division ring}. Therefore, A and A^ are regular and semi-simple.
Next, we shall show that they are amenable. Put S = [ 2 ® M , £ • M ]
a
j
otp ^.
ap
a
a
then S. = T S . Let e be an idempotent in S„ - 7TS ; a = T e , e € S ,
M ^ a
^
M a *
a
a
a*
a
a
0
Q
F
f
-2
e
.
= e
:
a
£®M
y
ag
a
-
. Then there exist a regular element a € S and a projection
j
»M , in
a
0
$
"t*
such that e = a
a
a
f a by ( 1 . U . 5 ) , (note
a a
*
J
a
may be regarded as the endomorphism ring of a vector space). Since f
is the projection in M _ , f sKts in A. Hence, so does e
_
fa
"a ~
if
_
regular, and e :M
Im f
2L> M .
since "a
Q
is
1
a
a
a
^
a
Therefore, so does "e, which implies that A (A ) is amenable. Thus, A (A )
f
is abelian and spectral by ( 1 . 3 . 2 ) . On the other hand,
is a minimal
object by ( 1 . U . 7 ) . Hence, A is completely reducible. Finally we shall
show that A satisfies the condition 3) in the definition of Grothendieck
categories. Let {A } be a directed set of subobjects in an object F and
35
Applications of Factor Categories . . .
B a subotject in F. Put C = U(Â B), then B =
K
(UA ) B
= (U A ) n ( C ^ B ) ="c ((\J^ r\B )
K
K
KJ
a
n
(V yA )
a n
a
o
a
o
i
C«B , since A is spectral.
°
since C i U ^ . We assume
B = D ^ 0 . From an exact sequence : £ •
K
—^W^A^
q
— 0
we obtain a monomorphism g:D — ^ Z $A^ such that fg = 1- , because "A
is spectral. Let D be a minimal sub-object in D. Then g ID is a column
n ^
finite matrix from the first part. Hence, Im (g|D ) Ç Z7 $ A
and so
° o
a.
1
l
_
n_
_
_
D S" U'A
Ç a for some 6 s K such that 8 > a. . Thus, D C~À f> B C C
c
ou
p
i
o p
and D S B , which is a contradiction. Therefore, ( h i ) B = 0 (A r)B).
o
o
K,a'
K a
i
0
v
36
N A
V
Applications of Factor Categories . . .
CHAPTER 2 . THE THEOREM OF
KRULL-REMAK-SCHMIDT-AZUMAYA.
In this chapter we shall prove the titled theorem as an application
of
(1.U.8).
2 . 1 . Azumaya theorem :
1
Let
{M }_ be a set of cinde. modules and M = £ $ M .
a I
I
LEMMA 2 . 1 . 1
[l] .-Let M and {M ) be as above and
x
be any element in S^.
S
= [M, ] . Lets.
M
Then for any finite subset
j}?^
°f
{M }_ , there exists a set {M.}? , of direct surmand of M such
a I
I i=1
9
J
that M =
T\ § M.
1 , $ M
1=1
qfiFta.}
J
and M . is isomorphic to M. via
i
a
Proof. -
Let e^ be the projection of M to M ^ . Then e^alM ^ and
e (l-a)e lM
1
(1-a) /or eacft i.
or
1
are in [M
a1
, M ] and 1
= ( e ^ + e^l-aje^l M
a 1
.
al
Since M ^
is cinde., either e^ae^ I M
b
6
or e^ (1-a)e^l M ^ is isomorphic :
1
M .
* b(M )
y M
, where b = a or (1-a). Hence, M b(H^) $ Ker e = *>(M ) • ¿ 1 •
• Repeating this argument on the
1
M
1
a
last decomposition, we obtain ( 2 . 1 . 1 ) .
LEMMA 2 . 1 . 2
[1*] .-Let J* be the ideal in § 1.4. Then J
non-zero idempotents.
37
1
does not contain
Applications of Factor Categories . . .
Proof. -
Let e be a non-zero
finite subset {M
idempotent
of {M )
Oil
T
in S^. Then there exists a
n
such that eM^ £
« M . 4 0 . We apply
1—1
Ot -L
1—1
n
( 2 . 1 . 1 ) to e and {M .}? ,
ai
. Then we can find a direct summand
i=1
• M.
^
I
of M such that M. = b.(M . ) , where b, = e or ( 1 - e ) . It is impossible that
1
1
ai
i
all b. are equal to ( 1 - e ) . Hence, e.ee . is isomorphic for some i, where
I
i ai
e
:M—* M
. e- : M — > M. are projections. Therefore, e £ J
'
I
1
by
I
(1.U.3).
n
LEMMA 2 . 1 . 3 . - let M = Yl ® N. and
I-I
N.
c.inde.. Then J
1
is tfce Jacobson
1
radical of S.,.
Proof. -
Let x = (x^j) *b i J * Then we note that 1 ~ x ^ is regular in
e
n
1
and that a sum of non isomorphisms of
*i
is not isomorphic. By the
i
above remark and ( 1 . U . 2 ) we can find regular matrices P,Q in
such that
P(1-X)Q = 1 ^ . Hence, X is quasi-regular.
We shall consider a similar lemma in a case of infinite sum in
the next section.
1
Now we can prove the Krull-Remak-Schmidt-Azumaya theorem.
THEOREM 2 . 1 . U
fl, 7 , 1 7 ] .
-
such that M = Z 9 M
I
Let {M }
T
, {N }
Ot 1
p o
= £ $ N
J
D
Q
T
.Then
I) There exists a one-to-one mapping $
38
be sets of c.inde. modules
of I onto J such that
Applications of Factor Categories . . .
Ma « N.4>(a)
/ x
for all acl
and hence* \ l\ = IJl», where
\l\ is the
1
1
J
cardinal of I.
1
IIJ For any finite subset I of I
there exists a one-to-one
%
of V into J suo/z tfcat 1YL
mapping
a
i-r
i£J'
f
/or alZ i€l' and
x
1
II'J For any finite subset I of J , there exists a one-to-one
1
N
mapping ij/' of 1 into J suoT? tfozt M^^? ^»(i) f
M =Z
%
• M.
IIIJ
TL
0 N
fil
a
i=1
a
1
^
i^ '
.
M' fee a direct summand of M, t^zen M
n
soffit YZ ® M . or /or any
or
m <°° M
1
1
is isomorphic to
contains a direct summand,
i
m
which is isomorphic to some V"* €) M . .
i-i
a i
Proof. -
, N } , v and J the
a
o [lid)
—
I) Let A be the induced category from {M
ideal in A defined in 6.1.U.
f
q
t
t
!
Then A/J = A is a Grothendieck and com­
pletely reducible category by ( 1 . ^ . 8 ) . Furthermore, we know from its
proof that M = £ $ M
j
ot
] L . Since M
=
j
and N
a
p
0
are minimal objects,
p
there exists a one-to-one mapping (J) of I onto J such that M ^ ^ N ^ ^ j ,
(note that we may use the similar argument in A to the ring theory, since
1
A is a good category). On the other hand, S ^ J
is equal to the Jacobson
—
a
radical. Hence, IM
N
implies M « N . , > as R-modules by the remark
a
( j ) ( a )
*
a
(J)(a)
3 in § 1 . 1 .
39
Applications of Factor Categories . . .
II) Put M
= r ®M . and let p : M + M
0
j
= Ker p $ £
i
^
be the projection. Then M =
0
1
©N^^j
since A is completely reducible, where
It is clear Im p = E «M . . Put N
= E
$ H ,
M
a
% and let i : N
^^( i)«
a
+ M
N
be
!
the inclusion. Then pi is isomorphic in A. Since N^e A^ , J o [NQ,NQ] is
equal to the radical of [NQ» J ^
( 2 - 1 - 3 ) . Hence, pi is isomorphic in
N
by the remark 3 in § 1 . 1 . Therefore, M = N
M = N
« M .. It is clear that M V N.,
« E
r
$ Ker p in
Q
and so
x in M_.
II ) The following argument is dual to that in the above. Put M- '= J2 ©M
0
a.
I
v
Since A is completely reducible,
Let p
1
H
?
= M ®
M
n
<BN
ftl
, where
:
Ker p' =
ET
(*). Let i
MJ
f
is in A
«Ng, , Im p = £
: MQ
1
j
N
1
and
M , & N
,, > .
M
Ip VOU;
t
A N ( I
$ ^ (ot!)
PI^Q* *
s
^
s o m o r
h
P i
!
C
$
J-r(l')
0
by
T
Q
andM = M ' e
c
M be the inclusion, then p i is isomorphic. Since
, p'i is isomorphic in M^. Therefore, M = M $Ker p
f
...(*).
* N / m , It is clear that
be the projection of M to N ' = E
u
1• + J
?
in
•
B
!
III) Let e be a projection of M to M . Since A is completely reducible,
Im e =
M
Q
=
j
* M
f
ot
t
, where M'
* M' » «> M
a
ot
Q
= ^7
are isomorphic to some M
0
p
in {M }_ . Put
a i
1=8
in Mp. Then from the definition of A,
40
Applications of Factor Categories . . .
we have the following R- homomorphisms : i : M ' + M
Q
p:M
*
M
N
'
— ^M
0
such that i is the inclusion
Q
M*">M
0
^ M and
S
p:M —> M-
1
0
f
is the projection and i'e = e. Since MQ £ A^, and pei is isomorphic in
A , so is pei in
;
M ' — ^
M —
0
Hence, Im e in
= M
1
M
...
(**).
contains Im ei , which is a direct summand of
n
M and isomorphic to X!
I M ' > . If I
i
f
is infinite, the above argument
gives the last part in III). We assume I
can take M J = M^. Hence, M
!
!
is finite. In this case, we
= Im e contains Im ei as R-direct summand
from (* *) • On the other hand, Im e = Im ei and hence, M
t^oo
ei by ( 2 . 1 . 2 ) , which is isomorphic to
$ M *.
F
is equal to Im
1
i=1
a
i
REMARK 1 . In the above proof, we used only an assumption "I is finite"
1
M
M
to obtain that J'rt [" » <}
i s
0
e
<iual
t o
t h e
radical of [M^MJ] for some
module M^. Hence, if we can show the above property with another assump­
tion, the proofs given above are still valid. We shall make use of this
fact in Chapter 3.
2 . 2 SEMI-T-NILPOTENT SYSTEM.
We shall
give, in this section, a new concept which is a genera­
lization of T-nilpotency defined by H. Bass [2]
.
Let [My] ^ be a set of modules (not necessarily c.inde.). Let A
be the induced category from { M \ and C an ideal in A. Take any countably
infinite subset
{M
a.
} of
{M } and a set of morphisms {f.'M^ + M^
,
a
1
a.
ot. .
41
1
1+1
r
1
Applications of Factor Categories . . .
f. C C}. If for any such sets and any element m in M , there exists a
~
d
natural number n (depending on the sets and m) such that
1
f
a
n n-l
f
•••*V
m
)
=
\
°>
with respect to
^
s
Let 1M^>
c
a
l
l
e
d
locally semi-T-nilpotent system
a
be a countable set of modules
such that
M. are isomorphic to some ones in { M } . If any such set and any set of
1
CL
morphisms f^ satisfy the above, we say {M^} a locally T-nilpotent system^
( [ 1 7 , 2 8 ] ) . If I is finite, we understand by the definition that {M }
is a locally semi-T-nilpotent
on any element m in M
system. If the above n does not depend
, we omit the word "locally". If every M
is fini-
Ot
Ot^
tely generated, we have this situation.
In this section, we give a principal lemma ( 2 . 2 . 3 ) , which we shall
frequently use later.
Let M = £ $
and describe End(M) =
by the ring of the column
summable matrices. We may assume I is well ordered. Let ai < 0t2 <
(or ai > a 2
• > a ) be in I and b
n
... a-) we denote
n _ 1
1
b
a
n
e [M
n
i i-1
b
a
b(a , a
<a
nVl
,M ] . Then by
i-1 i
... b
^
a
a
V 1 V 2
a
f o r
2 i
t h e
sake of simplicity.
LEMMA 2 . 2 . 1 (Konig graph theorem). - Let M , {M }j and £
Let f = ( b ) be in S ^ C . Put
QT
5
= {b(a , V l "
n
as above.
,
,
a
2
,
a
i
=
x )
>
for any n > 2 } . We assume 04^}^ is locally semi-T-nilpotent
system with respect to C. Then for any element x in M
T
b(a ,a ,..., a ) (X ) = 0 for almost all b in F .
n n*~ I
I
T
x
i
1
42
,
Applications of Factor Categories . . .
Proof. - Since ("b ) is column summable, there exists a finite subset
ox
T, of I such that b
1
OT
(x ) = 0 for all OCI-T,. Let 6 be in T,. Then the
1
T
1
subset T = {Y|b(y,6,T)'x ) 4 0 } of I is also finite. On the other hand,
2
T
{ M }_ is; locally semi-T-nilpotent and b C C , since C is an ideal,
a I
OT
—
r
Hence, ( 2 . 2 . 1 ) is clear from Konig graph theorem.
REMARK 2 . Let b(a ,a , .... a j be as above. Then for T < O
n' n - 1
* 1
b(o,a ,...,a ,T) is an element
I
2
T
in[M^,M^.
LEMMA 2 . 2 . 2 . - Let { M }_, M and C be as above. We assume {M } is
ot r
—
aI
T
e
n
locally semi-T-nilpotent with respect to £• Let (^ ) ^ ^
aT
such that b
a T
= 0 if o 4 T . Then (b
) is quasi-regular, (cf.
[ 33,261).
Proof. - Put B = ( b ) . Then each entry of the column of B
aT
n
consists of
00
some elements in F
S
M
t
. Hence, £ B has a meaning and is an element in
by ( 2 . 2 . 1 ) . Put A = E
n
B . Then (-A)B-B = - A. Hence, B is quasi-
regular ,
LEMMA 2 . 2 . 3
[19] (principal lemma) .Let (M )j be a set of modules and C
an ideal in the induced category from {M }. By S we denote
ot
ot
End(M^) . Suppose
1) C O S C j(s ) for ot£J.
a
A
2) If { )j, is a set of morphisms in £fl[.M,M^ such that { )JT
a
a
is summable, then Z a^ € C f\ TM ,M 1 .
a — L a* T
J
f
43
A
—
Applications of Factor Categories . . .
3) {M )j is a locally semi-T-nilpotent system with respect to £.
^en
C^S CJ(S ) .
M
Proof. - Let A
!
M
= (a
f
aT
!
)
be in £/0S and put A = (a ) = E-A , where E
M
aT
is the identity matix. We shall shew that A is regular in
lar argument to (2.1.3). Since Aa
1
S . Put b_< = a^-a.,
a
ai
ai 11
We shall define b
i) {b_ }
y
1
5
b
=
cx
aT
a
=
cx
_ y
s
1
a
a
» ao ^
s
r e
6
u l a r
n
for a > 1, then (b^, 1 is summable and b „ €C.
'
01 a
ai -
—
ax
y
T i
b
^
We defined {b^ }
a
1
w
>
ot
h
e
r
e
a
a
^ » t' t-1 '• * *' i ^
i>a
a
c f
x ' *'
a
^ -
Ren
iark 2).
1
t
with i) and ii). We suppose we have defined {b^} for
p<C. Then since every terms in (*) are defined, we can define y
Since Yl
^
for 0 > T with the following properties :
ax
+
n J
is ^ ^ ^ ^
is summable and b_ €C.
ai o
1
1
by the simi­
by (#) .
MT,CL
,...,aj a^ € CflS £J(S ) by (2.2.1) and 1.2 , y
' t1 ot ^ T
T
T
T T
is regular in S^ . Hence, we can define b
by ii). It is clear from
,
(2.2.1) and 2 that {b„ } is summable and b
OT
C = (c
) by setting
OT
=
00
b
a
A
,c
OT
T
0
€
M
f o r
M
T
= 0 for
»
OT
n
a
M
d
—
= 0 for 0 < T and
a* ' °r »
2
^ ^ [ ' l ^
i
summable and hence, C cS... Put D = CA = (d
C
€C . Now, we define
OT
c . = 1
a > T
'
T h e n
c
i s
c
°lumn
). First we shall show
OT
G>T .
OT
d
ax
=
I c a
p
ap px
= a + I
c a
ax < op
px
f
a
J
=a
ax
44
+
E
p<(ja>
£ ,/
b(a
o,... ,a ,p). %
T
2
Applications of Factor Categories . . .
b
V
f
t
OT
+
(
a
£
a
A
)
i
a
+
v
V J
t
+a_,)+ IT
a
a
t
a>a%>T
a a
b(ot" „ , . . . , a " ) a „T+a
( Z
t
t" a".
1
1
t"
-
V ""•"VSy
i
* „
Oa'^./t
b ( T a
„d „
t"
°V
a
i
T
a
1
„ )=y +*> y +Z
T
b
o
a
a>a"»>x
t"
&
a
V
... (**).
,T
It is clear from (**) d
= 0 for all x. If we use the transfinite
T+IT
induction on c,x, we can show d
= 0 if a>x from (**). Futhermore,
d
is regular in S . Put C. =
= £b(a,a ,.. . ,a. )a
UU
diag(d^
Xf
1
#
I
OL
+ a
^o
,d
OO
a
*• * aa ^»• • • ^
n
d K
O
I
~ E-C^CA^E-C^D. Then the entries of K,
which are in the diagonal or under the diagronal, are all zero and the
entries of upper the diagonal belong to £ by ii) and 2. Hence, K is quasiregular by 3 and (2.2.2), (which is a case of ai>a2>...>a ). Therefore,
n
CjCA is regular in S^. Again using (2.2.2), we know C is regular in S^.
Thus so is A. Therefore, C(\S 9 j{S ).
M
M
xt
yr
REMARK 3. - In the introduction we defined" take out property" of a module
M, which is the property II) in (2.1.U) without the assumption of the
1
finiteness of I . In that definition, we assumed that any kinds of decom­
positions of M should have the take out property. Now we fix a decomposi­
tion of M : M = T ® M^ , M
are c.inde.. We shall note that if this decom-
a
I
position has the take out property for any another decompositions
1
M = S ©Ng , then so do any kinds of decompositions of MrMsZTGM ,. Because,
J
K
a
let M = £ ®M = E«M» = ZT $ N . Then there exist a one-to-mapping 4> of
I
K
J
45
t
0
6
Applications of Factor Categories . . .
P u t
K onto I and a set of isomorphisms f ,:M^ •+
a
•
f
F
=
f
E a
| € S
M'
which is isomorphic. Hence, M = Z ©M = E ©F(N )< It we apply the
I
J
a
7
take out property for those decompositions, we obtain
M = £ «F(K., j€> 2T ®M
1
*
l-l<
1
( a )
. Therefore, M = F ( M ) = C ®u
_ 1
a
aeK'
46
*
$ Z «M\.
(
a
)
K-K-
a
'
Applications of Factor Categories . . .
CHAPTER 3 . SEMI-T-NILPOTENCY AND THE RADICAL
We have defined a (locally) semi-T-nilpotency for a set of modules
{M^}^ in Chapter 2 . In this chapter we study some relations "between a
semi-T-nilpotency of a set of c. inde. modules {M )j and the radical of
End(M), where M = Z 9M .
I
a
3 . 1 . EXCHANGE PROPERTY,
We shall define, in this section, the exchange property of a direct
summand of a modules, which is slightly weaker than the usual one (cf.L^] ).
DEFINITION.-Let N be an R-module and N a direct summand of M. We say N
has the a-exchange property in M if for any decomposition of M:M=£$T
I
with |l|$a , there exists always a new decomposition M = N © H©T' such
I
Y
}
Y
that
T^'CT^ (and hence, T
;
1
is a direct summand of T^ for all y 6 1 ) . If
N has the a-exchange property for any a , we say N has the exchange property
in M. If in the above, N has the a-exchange property whenever all T^ are
c.inde., we say N has the a-exchange property with respect to c.inde.
modules.
REMARKS 1 . It is clear from the definition that M has always the exchange
property in M.
2 . Suppose M = IT © N.. If N ,N have the a-exchange property in
i=1
M, then so does N^GN^ by
. However, the converse is not true.
p
1
47
Applications of Factor Categories • . .
Furthermore, even if neither
nor
has the a-exchange property in M,
it is possible that J^OT^ so does.
LEMMA 3 . 1 . 1 .
- Let {M } be a set of {cinde.) modules and M = £ ®M •
T
a
I
a
I
f
Suppose M satisfies the take out property for any subset I with
1
! 1 | <^Xo • Then { M ^ j is a locally semi-T-nilpotent system (with
f
respect to jJ ).
Proof. -
Let
be a subset of (M }j and
+
a set of
given morphisms. First we shall show that some of (f^) is not monomorphic.
1
Put M. = {m.+f.(m.)i m. €M.}cM.eM.
l
i l l '
1
1
1
+
1
<® M and M
1
= £
o
®M^ , where
-T%.
Y
l lo
Io = (1 , 2 , . . , ,n. . . ). Then it is clear that M ^ M ^ M ^ M ^ M ^ ® . . .®M
1
C
,
...(#)
=M '®M ®M ®M^®.. .Wo-*
1
2
3
We assume that all f^ are monomorphic and use the take out property for the
above decomposition. We take a subset I = (2,U,,..,2n,...). Then we obtain
!
from the take out property that
M=M
1
•««
..«M «* (M )EIP (M )E.. • • *
0
where ^ n ^ 2 n ^ ^
2
modules in M
Q
1
2n
£
M
01
o n e
°^
U
U
m 0 (
i l
u
e s
2 N
(M
2 N
)«
(**)
n
i the first decomposition except
has to be equal to some
§
3
®^2n^ 2n^ ^
some M
ua
^ ^ ^°
2
. From the above assumptions no one of {f^} is epimorphic
Kence, every M^
r "WV
S e<
2
r V"
M
^ 2m
?
*
WBSha11 Sh0V
V
e
h
a
d
S
O
m
e
(M,~). Therefore,
2m
2m
f, * * 2 n
2
i
S
u
c
h
. First we assume that we had ^
2k+i
h
a
2n
t
)
=
^,
< B M
M
^2i^ 2i^
2m
i
s
,
I
f
:
e (
*
u a l
t
o
(M )=M . . and ifu (M- )=M^.
0
2n
48
t
( M
2n
0
¿1+1
2m
2m
2J+1
Applications of Factor Categories • • .
M
,M
+M
for i < j. Then since M ^ ' is equal to some <l>2p^2p^ 2i+1 2i+1
l+M
f
2i+2 +
!
...+M . + M_.
is a direct sum from (**). We shall denote f f ....f
2j
2j+l
p p-i
q
0
c
w
"by 9(P5 l) for p>q. Let x ^ 0 "be in * i + 1
f
t h e n
2
= f
x
x +
2 i + 1
(x)
f
- 2i 1
( x )
+
€
f
f
- 2i 2 2i 1
+
{ x )
M
;
2 i +
€
W
+
(***)
+ (e(2j-1,2i+1)(x)+9(2j 2i+1)(x))C M
t
2 j
;e(2j,2i+D(x)eM
•
,
2j+1
which is a contradition to the above. Therefore, if Z ^ 2 n ^ 2 n ^ ^ ^ ** 2n' *
M
M
we should have only one ^2k^ 2k^
2i
M
-H
*V
p-1
e
M
P i + i Pn i
eM
+
P
2 1
Since ^2i+i *
1
s n c r t
2 1
1
> e
H
k>2i+1
wil
^
cil
*
s e <
l
u a l
t o
2i+1
V
o = IT ®
*
°
q-1
e
e
M
q
s o m e
M
n
M
f
8
2i+1 * T ^ *
/
•J « .
® im f . e £
k>2i+1
p
2 1
k
+1
1
*
epimorphic, we can show by the same argument to
(**#)
that M2i+2^"**' Therefore, some of (f^ has to be non-monomorphic. From
those arguments, we may assume there are infinite many of non-monomorphisms
f. among [f.\
Then all
1
. Let f. ;f. ,...,f. ,... be such a set. Put 6(i ^i~ >iv) Sv
1 2
n
are non-monomorphic. In order to show that (f^
is a locally
s
v
semi-T-nilpotent system, it is sufficient to show that so is [g^] . We
. Let x ^ 0 € Ker g^, then x € M * D M^* . When ve use the
1
put M^* •
k
M
above argument for (m^^J , we know from (**) that ^2n^ 2n*^ *
49
s n o t
e <
l
u a l
Applications of Factor Categories . . .
*. Therefore, 4 u (M. *) is equal to some M-* and
1
to any
2m+1
2n
K = M* 6M^ ®...®M
!
1
,
2
2n
¿011
1
(it is possible that some M * may not appear in
2m
o
this decomposition). Take x^O€M^ and use the formular (***) , then we
know that there exists some t such that 6(t,l)(x) = 0 . Therefore, { f ^
is a locally semi-T-nilpotent system.
We shall later make use of the following lemma and we can prove it
by the similar argument to the above and so we shall leave a proof to the
reader.
LEMMA 3 . 1 . 1 . - L e t {M }_ and { N } be sets of c.inde. modules. Put
a I
pJ
f
n
T
T = H GM^G £ ©Ng . We assume that £ «Ng
has the \ -exchange pro0
perty in T. Then for any countable subsets {M.} and {N.} of {M }
a
I
and {N«}, respectively and for any non-isomorphisms
pJ
"
f^:M^—^ £L, g^N^ — >
M
i+1'
a n
^ ^
r
an
y
X €
^ i > there exists m
sue?: tfcat g f . . .g-f- (x)=0.
m m
1 1
&
&
The following main theorem gives us an answer in a case where we
drop the assumption of finiteness in Azumaya' theorem (2.1.U).
THEOREM 3 . 1 . 2 [19,2U]
(MAIN THEOREM). - Let (M ^ be a set of c.inde.
modules and M = H $M
I
. Then the following statements are equivalent
a
DM
1
satisfies the take out property for any subset I and any
other decompositions (cf. 2 Remark 3 in Chapter 3).
2) Every direct summand of M
50
has the exchange property in M.
Applications of Factor Categories . . ,
3) Every direct summand of M has the exchange property in M with
respect to c.inde. modules.
4) { M ^ } ^ is a locally semi-T-nilpotent system with respect to J
defined in §1.4.
f
5) J n End(M) is equal to the Jacobson radical of End(M).
Proof. -
1)
k) It is clear from ( 3 . 1 . 1 ) .
k) ~ * 5) Since S / J ' n S
M
is semi-simple by ( 1 . U . 8 ) , S^n J/ 2J(Sjj), where
M
Sy = End(M). We shall prove the converse inclusion from ( 2 . 2 . 3 ) . The first
condition in ( 2 . 2 . 3 ) is clear for S . Let {a^} be a set of element in
f
J ^fM ,M*] such that (a^ is summable. Put a = l a . , If
, then
a<» J'n | M ,M 1 , If M
, we can show by the same argument in the proof
—
L
A
a ' T
O
T
of ( 1 . U . 2 ) that a is not isomorphic. Hence, a £ J r\ [M
1
, which is the
second condition in ( 2 . 2 . 3 ) . The third one is equal to k). Hence,
J ' 1 S C J ( S ) by
M
(2.2.3).
M
5) -—* 1 ) Let M' = ¿ 7 0M and e the projection of M to M . It is clear by
F
I
?
Y
( 1 . U . 3 ) that ( J o S ) O S
f
u
u l
= J'r\$
On the other hand, it is well known
that eS e = S , and J(S^ ) = eJ(S )e. Hence, (
J
M
M
f
M
S
M
, ) - J ' ^ S ^ , which
guarantees 1 ) by Remark 1 in § 2 . 1 .
2 ) — > 3 ) It is clear from the definition.
3) —* 1 )
3) implies k) by ( 3 . 1 . 1 )
and hence, implies 1 ) .
1) —^> 2) In order to show this, we need the following proposition.
If we use it, the proof is clear.
51
1
Applications of Factor Categories . . .
PROPOSITION 3 . 1 . 3 - - Let {M^} and M be as in (3.1.2). Then the following
statements are qeuivalent.
1) The property III in the introduction ; every direct summand
of M is a direct sum of cinde. modules M
T
such that M
ot
isomorphic to some
f
are
ot
in {M^}^, is true.
2) For any idempotents e,f in
we have
f
eS * fS if and only if eS^/ei J»0 S ) * fS^/t(J n t^).
y
M
M
Proo/. - 1 ) — ? 2 ) Put S = S /J /1S , eM =I®M .and fM =I®M^ . We
,
M
,
M
M
a
rt
Assume e S ^ fS^. Then Im ecrlm f in A, where A is the category in (1.U.8).
Hence, since Im e = £ ®M
f
1
and Im f = £ « M ',,, M
ot
ot
f
f
is isomorphic to
ot
some M'\, and vice versa by ( 1 . 1 + . 8 ) . Therefore, eM^fM, which implies
e S
M^
2)—>>
F
S
W
1 ) Let M
f
be a direct summand of M and e the projection. We showed
in the proof of ( 2 . 1 . U ) that
fM
there exists an idempotent f in
such that
= £ m ; 1 C I and e S . f S . Hence, eS.,fS.„ implies e M ^ fM.
•p
6
M M
M M
1
x
M
COROLLARY 3 . 1 . ^ [ ? ] . - Let {M^} be as in (3.1.2). If one of the conditions
in (3.1.2) is satisfied, then the property III is true for M.
REMARKS 1 . We can replace 2) and 3) in ( 3 . 1 . 2 ) by the Xo-exchange property
by virtue of
(3.1.1).
2 . Let Z be the ring of integers and p a prime. Then {Z/p }?
1
so
is not a semi-T-nilpotent system. Hence, M = 21 $Z/p does not satisfy any
i=1
statements in ( 3 . 1 . 2 ) . However, M satisfies the property III (see § k.2).
52
Applications of Factor Categories • • .
3 . Let ÍM J ^
e a
s e t
of indecomposable modules with finite composition
lengthes w&ich do not exceed a fixed natural number n. Then i^ ^j ^
s
a
a T-nilpotent system with respect to J
?
(see [lTl).
U. Let K be a field and R the ring of lower tri-angular matrices with
infinite degree. Put M = 2 € e..R, where e.. are matrix units in R. Then
°
1 1 *
li
i
{e^RJis not a semi-T-nilpotent system, but M satisfies the property III
(see § U.2).
5. Let R be the ring of upper tri-angular matrices. Then {e^RJ is a
T-nilpotent system.
3 . 2 . DENSE SUBMODULES.
In this section we shall give a special answer to the property
III. Let {M }_ be a set of c.inde.modules and M = £ $ M . By A we denote
a I
j
a
~~
the induced category from
Let J* be the ideal in A defined in §
1
We denote A/J by A.
DEFINITION. - Let M and N be in A such that N is a submodule in M,
i:N —*M inclusion. If i is isomorphic in A, i.e. N = M; N is called a
dense submodule in M, (note that if N is a submodule of M which is a
direct sum of c.inde. modules and i is isomorphic in £, then N e A , where
£ is the induced category from all c.inde.modules)•
=
NOTATION.-Let e be an idempotent in S^ End(M). Then M = eM*(l-e)M in N^.
We do not know whether eM £ A or not, however we shall denote Im e in A
by eM for the sake of conveniency. It is clear that if eM€A, Im e = eM
in A. We note that even if f(M)
€ S
is in A for some ^ » Im f is not
M
53
Applications of Factor Categories . . ,
equal to f(M) in general.
PROPOSITION 3 . 2 . 1 . - Every dense submodule of M is isomorphic to M.
, M=crN as R-modules by ( 1 .U.8),
Proof. - Since M = Z7 $ M = If = E I II
I
where N^'s are c.inde. modules.
a
Y
PROPOSITION 3 . 2 . 2 . - Let M and P in A and M => P in A. Tten t?zere exists a
submodule P
Q
in M which satisfies the followings :
k i.e. V = 2 e M .
°
J
For ant/ finite subset J' a/ J Z , ©M* , is a direct summand of M.
J
J/* {M }
is a locally semi-T-nilpotent system with respect to J\
1) P
1
is in
1
a
1
a
T
then P is a direct summand of M.
o
*
.?J P ^ P as R-77?oduZ^s.
q
Furthermore, if e
can find such P
q
is an idempotent in S^ and P = Im e, then we
in Im e in M^.
Since A is completely reducible by ( 1 . 1 + . 8 ) , there exist R-homo-
Proof. -
—=•> P such that pi = 1 — . Let P = Z © P
K
are c.inde.. For a subset K of K w e denote the injection :
morphisms i: P —* M and p:M
P
f
P^
P
K
!
;
Y
=
^
P
^ y~^
vely : P-, ^
P
a n d
—
P ,
K
t h e
P Jection : P — * P , by i
ro
K
P « — * M. Then p , ,p . , = 1
P
K'
v
>ir
K
l l K
D
R
If either K' is finite or
R |
) by (3.1.2). Hence, p„,pii ,
K
K
is R-isomorphic. Therefore, ii , is monomorphic in
54
and p . , respecti-
P
{P }„, is semi-T-nilpotent, S_ r\ J' = JCS^
Y K
P,
P
K
R f
for every finite
Applications of Factor Categories • . •
subset K
Then P
f
of K, which means i is monomorphic in M^. Put P
satisfies 1 ) ~ 3 ) .
q
= Im i in M^.
q
Suppose Im e = P. Then M = P • (l-e)M and
hence, pei = pi. Put P^ = Im ei in Mp. From the above argument, we know
that P
q
satisfies the all requirment i n ( 3 . 2 . 2 ) .
REMARK 6.
n
Let N = £
8 M.
be a submodule of M via the inclusion i N
1
i=1
Then we know by the above proof that i^ is monomorphic in A if and only
if N is a direct summand of M.
1
LEMMA 3 . 2 . 3 T ] ' ~
L
e
t
M
a
n
d
J
f
be as above. Then for any
F£ J'n S ,
M
1^-f is monomorphic.
Proof. - Suppose Ker (1-f) ^ 0 . Then there exists a finite subset I
such that Ker 0 - f ) n H
summand s {M
for each
!
®M
1
of I
? o. By ( 2 . 1 . 1 ) we obtain a set of direct
. J . , such that M = Z
}
<p{OL ) I*
,
$ M
f
. X $ ¿7
$N
CT>(a )
!
a
and M ,*M./
a
${a j
1
M
a €I' via either f or ( 1 - f ) . However, f is in J' and hence, we
?
must obtain those isomorphisms by ( 1 - f ) , which is a contradiction. Therefore, Ker (1-f) = 0 .
We shall give criteria £ or submodules to be dense.
THEOREM 3.2.U. - i t {M } be a set of c.inde. modules. A the induced
e
T
J
a I
category from {M }_
and J
1
3
—
the usual ideal in A. Let N be in A.
i.e. N = £ 8
and a submodule of M via the inclusion i^rN-^M.
J
Then the followings are -equivalent.
55
Applications of Factor Categories . . .
1) N
is a dense submodule of M.
f
2) i
is monomorphio in A/J and for any direct summand P o/M,
there exists a finite subset J* of J such that P^Nj, / 0 or
P © N
T t
is ne»* a direct summand of M, where N
T I
= £
N
$ ^f
1
is monomorphio and N contains Im (1-f) in M ^ /br some feJ .
T
Hence, Im (1-f) is a dense submodule in M for all f€ J . Furthermore, the above N „ is a direct summand of M i/ .either J" is
finite or {N »} „ is a semi-T-nilpotent system.
Proof. -
2 ) Since P contains a direct summand of M which is c.inde.
1)
by (2.1.U), we may assume P is c.inde.. Furthermore, since A is a
Grothendieck category and P is minimal in X, Pc £ $N , = N , for some
ji
Y
finite subset J of J. Suppose PON^, = 0 and P Q Nj, is a direct summand
T
!
of M ; M = P ® N «M . Let i : P$N — ^ M be the inclusion. Then
J
O
<j
TI
Im i = P $ N
T I
T?
, which is a contradiction. Hence, P$N
T?
is not a direct
summand of M.
2 ) = = ^ 1) We assume that i
is monomorphic and M ^ N. Then there exists
a minimal object M^ such that M^ON = 0 . Hence, for any finite subset J'
= 0 . Therefore, M^ $ N , is a direct summand of M by Remark 6
of J M ^ N
T
(take first a formal direct sum M^ 9 Nj, and consider a natural mapping
from M • N , to M u N
T
J
Ct
T I
CM).
d
1 ) ^ 3) Since ijj is isomorphic, there exists an R-homomorphism JC[M,N1
such that ijjj = 1 . Then f =
M
c J
56
f
and Im (1-f) in M ^ Ç Im i = N.
Applications
of Factor Categories • • .
3) = ^ 1 ) Since 1-f is monomorphic by ( 3 . 2 . 3 ) , Im (1-f) in
Put 1-f:M
( l
f t
1
F
is in A.
F
" > ' N'—L^M. Then
morphic in A, since ( 1 - f )
= N
= T-f = i ( W ) . Hence, I is iso-
is isomorphic in M^. Therefore, Im ( 1 - f )
1
is a
dense submodule in M. Since i^ is monomorphic and N ?Im ( 1 - f ) , N is
also dense.
The remaining part is clear from Remark 6 and ( 3 . 1 . 2 ) .
REMARK 7 . - In general, we have many dense submodules P in M = ZT ©M^ ,
i=1
n
for instance such as Po 5Z ®M. = 0 for some n < oo or PnM. # 0 for all
i-i
1
i (see [ 1 8 ] ).
In the above we showed that if J
1
is a finite set, then Nj, is a
direct summand of M for a dense submodule N. We generalize this property
as follows :
DEFINITION.^Let A ? B be R-modules and B = £ « B . If for any finite
J
subset J' of J, H ® B is a direct summand of A, we call B a locally
J'
direct summand of A (with respect to the decomposition B = £ © B ).
J
We note that if all B^ are infective, B is always a locally direct
summand of A. We shall use this fact in Chapter 6 . In general B = ^ ©B
I
is a locally direct summand of I B
I
T
Y
T
T
ì
THEOREM 3 . 2 . 5 .
- Let {M } be a set of cinde.modules and M =
$ M•
I
u i
Then the following statements are equivalent.
1)
{M } is a locally semi-T-nilpotent system with respect to J'
2) Every dense submodules coincide with M.
2) Every locally direct summand M of M with TMpect to W
!
=Z « T
V
57
Applications of Factor Categories . . .
with any cardinal I K ' is a direct summand of M.
4) Z) is true for decomposition with | K |^Xo •
5) 4) is true whenever all
S
J
S
s
6) ^ / ( ) ^
a
regular ring in the sense of Von Neumann and
M
S
J
s
every idempotent in ^/ ( ^)
Proof. - 1 )
are c.inde. modules.
^
s
lifted to S^.
2) Every dense submodule N of M is a direct summand of M
by the last part of ( 3 . 2 . U ) . Hence, N = M by ( 2 . 1 . 2 ) .
2) « = ^ 1 ) Since Im (1-f) is dense in M for f€ J ^ S
f /
2 ) . Hence, J'n S
1) w
M
Q J(S^)
9
M I
1-f is regular by
, which implies 1 ) from ( 3 . 1 . 2 ) .
3) Every direct summand of M is a direct sum of c.inde.modules by
( 3 . 1 . * 0 . The assumption of locally direct summand implies that £ ©T
^
K
a subobject of M via i^
f
!
, where i^. : M — > M inclusion. Hence, M
is
A
f
is a
lirect summand by Remark 1 in § 2 . 1 and ( 3 . 1 . 2 ) .
3) ~=s> U)
5) They are clear.
5)=-*l) We shall recall the proof of ( 3 . 1 . 1 ) . Let {Mj}" be a countable
subset of {M^} and { f ^ : M ^ — ^ ¿ + 1 ^
defined the submodule M
1
a
Si
v e n
s e
^ °^ morphisms in J/. We
f
= M, «M '6 ... in M. Since M
1
2
M
f
1
m
=
n n+1
n+1
1^ ® M- for any n, M is a locally direct summand of M. Hence, M is a
i=1
direct summand of M and hence, so is in M = 12 ® M.. Since M - im (1-f)
?
!
1
1
in K^, M
!
e
is a dense submodule of M , where f - S ~iiy|**i
°
i=1
matrix units in S^ . Hence, M = M . If we use the formula (***)
of ( 3 . 1 . 1 ) , then we know that
l s
58
a
e
»- •
s
Q X e
1 J
in the proof
locally semi-T-nilpotent system.
Applications of Factor Categories • . .
1 ) ~ ^ 6) Since J O S * J(S ) "by ( 3 . 1 . 2 ) and A/J is a regular ring by
—
M
M
—
—p —
( 1 . U . 8 ) , so is S /J(S.J. Let f es„ such that f = f. Then there exists
M M
M
f
1
w
W
x/r
a direct summand M^ of M such that M^ = im f by ( 3 . 2 . 2 ) . Let e be the
projection of M to M . Since Im f = Im e, Im ("l-f) ^ I m (1-e) in"^.
<1
such that f = "a ^ea by
Hence, there exists a regular element a in
(1.U.U). Since J*r> S = J(S ), a is regular in S and hence, a ea is a
1
M
M
M
idempotent.
6)«^1)
J'n
S
^ ( S ) by ( 1 . U . 8 ) . Since S /J(S ) is regular,
J
M
M
M
f
M
,
(J 0 S )/J(S ) contains a non-zero idempotent if J n S / J ( S ) . Then this
M
M
M
M
!
idempotent is lifted to S^ by 6) and hence it is in J / l S , which contra­
M
dicts
(2.1.2).
COROLLARY 3 . 2 . 6 . - Let R be a local ring with T-nilpotent radical J(R)
and S ^he ring of column finite matrices over R with any degree.
Then every idempotent in S/J(S) is lifted to S.
Proof. - Put M = £ © R, then S^End^M).
The following theorem is some generalization of (3.2.U) and is a
special answer to the property III.
THEOREM 3 . 2 . 7 . - Let {M } , M and A be as in (3.2.4). Let M = £ * N .
a 1
j
»
where
may not be in A. Then there exists a set of submodules
{P } of M as follows :
Y
1) N 2 P and P € A.
T
J
y
y
y
2) 2? ©P^ is a dense submodule in M.
59
Applications of Factor Categories . . .
Proof. -
(note that 11^ is regarded
Let II be the projection of M to N^
as an element in [M,M] . It is clear that
1
= 211
y
l f
M
. Let M „ be an element in { M }. For any non-zero element m. in
1
a
1
J
J
M
we have N (m ) = 0 for all
1
is a summable set and
1
J-J', where J* is a finite subset of J.
v
Hence, II
S J for all Y J - J K We shall express 11^ as matrices (x^)
in § 1 . 4 . Since {II } is summable, so is {x } for any a $. It is clear
F
e
Y
T
T
yo
Y
II |M- = (x )
Y' 1
ai a € I_. Therefore,
Then M- = I m T j M C I m ( Z
1
'
^
Hence, M =
d
OCp
1
jt
E II |M € J
j_ y
"
1
1
(see the proof of ( 1 . U . 2 ) ) .
J t
R |M. + ( C
Y •
IllMj = Im(2Tl |M )C ^
Y
J
Y
jt
v
!
1
Im II . On the other hand, there exists a set
fPyJj °f
submodule in N
Y
such that P € A and P = Im II . It is clear that
Y Y
Y
i) Im IK = £1
5eK
6€K
© Im IK for any finite subset K of J, and so
0
M
=
E 9 Im
iL
6
J
Iml.
Y
a
0
= £ © P .
J
Y
We shall call such P^ a dense submodule inN •
The following proposition shows that dense submodules in
maximal submodules in
PROPOSITION
N
f
3.2.8.
- Lei
a r
e
up to isomorphism in some senses.
M
be as above and N a direct summand of
be a dense submodule in N and T
is a locally direct summand of N, T
M.
Let
a submodue of N and in A. If T
is isomorphic to a direct
1
summand of N . Every countably generated ^-submodule of N is iso­
1
morphic to some submodule of N .
Proof. - We leave the proof to the reader (cf. ( U . 2 . 1 ) ) .
60
Applications of Factor Categories . . .
CHAPTER 4. THE EXCHANGE PROPERTY
Let {M^-j. "be a set of c.inde. modules and M = 2 ®M as before.
a
In chapter 3 we have considered a case where every direct summand of M has
the exchange property in M. We shall concentrate, in this chapter , in a
direct summand of M which has the exchange property in M.
U . 1 . SEMI-T-NILPOTENCY AND THE EXCHANGE PROPERTY.
Let M be as above> A the induced category from tM^}^ and A = A/j
1
as before. It is clear that if a direct summand N of M has the exchange
property in M, then N 6 A.
PROPOSITION
Let M = N
1
• N . If either N
2
1
is a finitely generated
R-module or its dense submodule is a direct sum of c.inde. modules
1
1
{M ^.}^ such that {M ^.}^. is a locally semi-T-nilpotent system^
then Tl GA.
n
Proof. - If N
is finitely generated, N. is contained is some
®M Z 9 M.
1=1
n
Hence, N.. is a direct summand of 2T •
i=1
III. If a dense submodule N
N
1
= N
1
f
M
1
• Therefore, N^€A by (2.1.U),
i
of N is of form in the assumption, then
by (3.2.U).
The following proposition is true in a general case (see D * f 3 8 j ) ,
however we shall prove it by virtue of a structure of A.
61
Applications of Factor Categories . . .
PROPOSITION U . 1 . 2 0*>38] . - Let M be as before. If M = N $ N and
n
N = y
9 M
,M
' s are c.inde. tften N. has the exchange
1
a.
a.
•
1
i=1
i
i
1
2
57
property in M.
Proof. - Let M = 23 • Q
dense submodule
N
®N
= E
I
P
® Q
1
a
be any decomposition. Then each Q
contains a
a
ji
, P„ =
= 2
I
1
2
J
p
YL
p
• „
$ p
j
(
»J
s
a r e
c.inde.. Then M =
Notation in § 3 . 2 ) . Since N
see
=
«M
i=1 i
a
a
-
m
-
—
N. is contained in a co-product of finite many of P
>say T2 2-]$ P. - ,
j
i-1 J!
m i=1
where J . is a finite subset of J.. Hence, P =X7 T Z $ P- • contains a
i=1
J
1
1 J
f
1
1
f
direct summand N' such that N ^ ^ N ^ in
N
it
»
P£A«
,P =
N e £ £
IP..
1t
and
; J-"^V
1
1 J
= N^ by ( 3 . 2 . 2 ) . Since
by ( 3 . 1 . 2 ) and ( 2 . 1 . 3 ) .
11
i j
^
i
Furthermore, P is a direct summand of M by ( 3 . 2 . 2 ) . Since Q. z> 22 • P - n
J.»
r-r
m
i
Q. = h
I P . . Ift. , Hence, M = N / « 1 7 E «P.. • 21 «Q. « 2 7 • Q . Let
J«.
i J."
i-1
a^i
I
I
I
1
1 J
1
1
1 J
1
1
p be the projection of M to N^
"p|N = p J n ^ and p i
1
1 J
1
a
in the above decomposition. Since
. Since N^A^ , pi^ is isomorphic in
1
1
m
1
e<
e
( 2 . 1 . 3 ) . Hence, M = N^Ker p « N • £! ( IT • ^\ j V ^ ^
0^«
1
N
= ^
1
= N* ,
by
1
i-1
J \
We note that if I is finite, we may regard i^^j
T-nilpotent system, (see § 3 . 2 ) .
62
a^i
as a locally semi-
,
Applications of Factor Categories . . .
THEOREM U.I . 3 . - Let
be a set of c.inde.modules and M » Z « M = N . . « N .
a
e M
Suppose N = Z
1
f
j i
I
, ;M
1
f
. are c.inde. If {M
Ot
01
0C
semC-T-nilpotent system^ N^ and
Proof. - First, we shall show that
t
2
} _ . ie a locally
x
have the exchange property in M.
has the exchange property in M.
Let M = £ $ Q "be any decomposition. By ( 3 . 2 . 7 ) each
contains a dense
J
submodule P - £ 9 P . . Since A is a completely reducible and
ot
,j,
oti
a
a
Grothendieck category by ( 1 . U . 8 ) , we have
M = N 0S
H
© P . , , where T' C T
J,a T •
a
... 1 ) .
0
2
a i
It is clear that N,1<fc Z Z
a
a
9a iP .. . Hence a,i{P
JT . , }
j t
<5t
T
T-nilpotent system by the assumption. Put p
Tm l
is a locally semi-
be the projection of M to
1
with Ker p
= N . From 1 ) we know that p | Z Z, • "P • i is isomorphic
1
1 J T
N
P
N
N
a i
d
t
in A. Hence, p.. I £ £ $ P . , is isomorphic in l/L by ( 3 . 1 . 2 ) . Therefore,
1 J T'
^
a
M = Z Z
P ., • Ker p„ =
Z $ p
$
i
£ $p
c o
j
>
ai
^
J T
ai
2
^
ai a i
Z
T
1
T
1
N
S
n c e
f
a
has the exchange property in M. Next, we shall show that N^ has the
exchange property in M. From the similar argument to 1 ) we have a dense
submodule P = P' § P " in
a
a
a
ot
« I « P'
M =1
j
1
"N
®
j
i»
...
such that
2 ) and
a
...
3).
a
63
Applications of Factor Categories . . .
f
Since M = r ©~P © Z © P" , there exists p€JM,E № " 1 in JL such that
_
ot
_
ot
ot
—k
J
O
u
Ker p in A = I © P
f
and J?|M is the projection of M to £ © P
a
n
a
... h).
From 3) and ( 3 . 1 . 2 ) we obtain M = ZT © P
© Ker p and hence, Q =
j
a
a
M
P"
© (Ker p/)Q )• Then M = Z « Q
= £ $ P"
© Z ©(Ker p
Q )=
0
Z" ©P" ©Ker p. Hence,
_
ot
1
J
Ker p = Z <B(Ker p Q )
J
n
... 5 ) .
a
From 2 ) and k) Ker p
n
N
= 0 and jp(M) = p(Nj) =Z ®P" . On the other hand,
1
a
from 3) we know that p { i s isomorphic in Mp. Hence, M =
N
1
©I© (Ker pn Q ) by
a
© Ker p =
5).
The following theorem is a generalization of ( 3 . 1 . 2 ^ 2 ) and 5 ) .
THEOREM U.1.U. - Let M
and iM } be as in (4.1.3) and M = N^Ng. Let f
f
be the projection of M to N . 2%£rc fj f = fJf if and only if every
1
direct summand of
N
a£so has the exchange property in M, where J
2
i
has the exchange property in M. In that case
=
f
=
S^ and
j ( ^ ) .
?
Proof. - We assume fJ f = fjf. Since A is completely reducible, there
exists a subset K of I such that Im f <y 2f • M » M~. Let e be a projection
K
of M to M . Then fS /fJ ^ eS /eJ . Hence, there exist a£eS f, b € fS^e
f
K
such that basf
M
f
M
M
f
f
fjf, which is equal to the radical S^
morphism in S^
f
and ab^e (mod J ) . Put f-ba = n € J . Then n = fnf£fJ f =
:K
1
= fM ^
= End(N ). Therefore, ba is an auto1
b
eM — • N . Then eM = a(fM)©Ker b in Mp .
1
64
Applications of Factor Categories
•
f
On the other hand, since ah s e (mod J ) , bJeM^>M is monomorphic (note
eM£A). By considering a dense submodule
of Ker b, we know Ker b = 0 in A.
Therefore, Ker b = 0 by (2.1.2) and eM-^rfM in P^. Since fJ'f=fJf, {
M
a
^
K
is a locally semi-T-nilpotent syfcem by (3.1.2). Hence, every direct
summand of N^ = fMfe A) has the exchange property in M by (U.1.3)• Conver­
1
sely, we assume that every direct summand
of
has the exchange pro­
1
perty in M. Then N = Z © T ; T are cinde. and N.. has the exchange
K
property in N.. Hence, (T
is a semi-T-nilpotent by (3.1.1). Therefore,
i
y rl
fj f = fJf by (1.U.3) and (3.1.2). The remaining part is clear from (U.U3).
Y
Y
f
COROLLARY H.1.5. - Let M and .N be as in (4.1.3). We suppose that for
i
every monomorphism g in S^ Im g is a direct summand of N i.e.
2 ^
1
gS
= eS„ and e = e . Then every direct surrmand of N- has the
exchange property in M. Especially, if N is quasi-infective, N^
1
satisfies the condition.
Proof. - Let f be the projection of M to N and e£fJ f. Then (1-a) is
f
1
monomorphic by (3.2.3). Futhermore, (l-a)JNp = 1
and Im (1-a) =
2
Im ((1-a)| N^ )*N . From the assumption, Im ((l-a)j N ^ is a direct summand
2
of N^ and hence, Im (1-a) is a direct summand of M. On the other hand,
Im(l-a) is a dense submodule of M by (3.2.U). Therefore, Im (1-a) = M
and so Im (O-a))^) = N . Hence, a is quasi-regular in S
1
It is clear fJfCfJ'f, since Jc J". Now we assume N
1
N
and fj'fcfjf.
is quasi-injective and
g is a monomorphism in S^ . Then we have a commutative diagram :
65
Applications of Factor Categories . • .
> Im g
0
g
— »
N
1
/
-i
e
k''
Since g
1
is epimorphic,
= im g © Ker 6 .
Faith and Walker [9] proved the above corollary and Warfield [39]
did in a more general case, where
is injective. Fuchs £ 1 2 ] generalized
£39] in a case of quasi-injective modules. Kahlon [25] and Ymagata [k6\
studied the corollary when all
As we see above, the
are infective.
locally semi-T-nilpotency of a submodule N
guarantees the exchange property in M (more strongly for all direct
summands of N ) . However, the converse is not true, for example M itself
has the exchange property in M, but its direct summands do not. Of course
this is a special example.
oo
Let Z be the ring
.
of integers and p. primes. Put M = £ €Z/p
i=1
1
1
oO
^
00
^
$ Z $Z/p , (p #P ). Since N = H § Z/p is the set of all p^primary
i=1
i=1
2
1
1
2
1
'
O
O
has the exchange property in M, but fZ/p^j ^ is not semi-T-nilpotent.
This example is similar to the first case. Let M s^GZ/p
has the exchange property in M if and only if either
1
» N^QN,^ Then
or N
2
is
isomorphic to a finite direct sum of {z/p } , (see ( U . 1 . 7 ) ) . Hence, in
1
this case either
or
must have the property of semi-T-nilpotency.
In the following we shall study those situations (I do not know
66
Applications of Factor Categories . . •
whether the concepts of the exchange property and semi-T-nilpotency are
equivalent, except special cases).
Let M « Z • M he as before and M = N.GNp. We noted that if H
I
has the exchange property in M, then N^C A.
1
a
PROPOSITION U.1.6. -Let M,
I
I
m(i)
J(i) *B
Y
be as above. We assume that N =
i
M
e
* vhere ^ ^ B °*
indue, and M( 1 ) g*M( 1 ) tf**
Y
y
y
Y B
y
M ( 2 ) i ^ M(2) g« owd M(i) gjk M ( j ) , g , t /
Yg
Y
we assume that if Q$|j(2) | <
00
Y * Y .Furthermore,
; I J( 1 ) | $ I J(2) |. 2%en H
y
y
1
to
and onZy t / {M(1) g} is a
the (Xo")exchange property in M
locally semi-T-nilpotent system with respect to J!
ft
Proof. - If" part is clear from (1.1.3). Conversely, let (M(1)
g
4
be a subset of {M(0 -} and {f.lH^L A
YB
il
}°°
i i=1
M( 1)
and f.€J) .
i+1 i+1
"
0
Y
From the assumption, we may assume that J(2)
Y
B
1
± 0 for all i and that if
i
|j(l)
j = oo , |J(2) | = oo . in order to show that {f.} is a semi-Ti
i
nilpotent, we may change f . by suitable g . :M(2)
—* M( 1)
^J
2i
2i
Y ^
Y
1
0
0
a
1
ft
from the above assumption. Then we obtain the proposition from (3.1.0.
If |j(2)
00
|=
for all Y» the assumption is satisfied.
PROPOSITION 1*.1.T. - Let {M }°° be a set of c.inde. modules such that
is monomorphic but not isomorphic to M ^ for all i.
CO
1) Let M = E « M = N^Ng. Then N has the (xo~)exchange
+1
i
1
67
Applications of Factor Categories • • .
property in M if and only if
or N
2
is a direct sum of c.inde.
modules [M^ijj which is locally semi-T-nilpotent (in this case
N.. or
is a finite direct sum).
2) Furthermore we assume that any of
is itself a locally T-
s
nilpotent system and M = Z ©T
^
;T
Ot
is isomorphic to some M..
1
Ot
Then we have the same statement as in 1).
Proof. - 1 ) If N^ and N
are infinite directsums of c.inde. modules, it
( 3 . 1 . 1 ' ) . We can prove it similarly to 1 ) and (U.1.6) and
contradicts
we leave
2
it to the reader.
Contrary to the assumption in ( U . 1 . 7 ) we have
PROPOSITION U . 1 . 8 . -
Let M « % © M and M
isomorphic to a fixed c.inde.
a
for all a. Let M = N^©N . Then N^ has the exchange
I
module
2
property in M if and only if N^ is a direct sum of c.inde.modules
{M ,} which is a locally semi-T-nilpotent system.
ot
J
We leave the proof to the reader.
h.2.
THE PROPERTY III.
We shall study the property III in the introduction, namely let
M = ^ © M
A
be in A , then every direct summand of M is in A. Whether the
property III is true for any M in A or not is still an open problem. If
{M }
a
is a locally semi-T-nilpotent system, this property is true by ( 3 . 1 . 2 ) .
We shall give the combined result (k.2.5)
68
of [38] and(*2U] .
Applications of Factor Categories . . .
LEMMA U . 2 . 1 . - Let M = I $ M^ = R^K^ be as before. For any x in N^
there exists a direct summand N
of N, sucft that x^N
o
and N C A ,
o
1
o
—
Proof. - It is clear that there exists a finite subset J of I such that
x6M = \
. Since M has the exchange property in M by ( U . 1 . 2 ) ,
M = M ^ N / e N ' where N.'CN.. Put N. = N.o(M ®N. )
Then xcN-"
J 1
2 '
1 - 1
1
1
J j
1
t
!f
and M = z «(N i eN i ). Hence, M
^ WSj" and so N^ C A by ( 2 . 1 .U).
i=1
i=1
T
T
11
f
1
0
T
,f
J < 5 r
1
1
COROLLARY h.2.2.
1
- Let M = N ^ N ^ be as above. If
is countably generated,
l^e A.
Proof. - We can prove it by an induction from ( U . 2 . 1 ) .
LEMMA U . 2 . 3 C ^]' ~ Let M be a direct sum of countably generated R-modules.
2
Then every direct summand of M
is also a direct sum of countably
generated ^-modules.
1
See [ 2 6 ] or [3 *] for the proof.
LEMMA k.2.h [ U,38.] - Let M = Z ©M^ and let all
be countably generated
and c.inde.modules. Then the property III is true fvr M.
Proof. - It is clear form U . 2 . 2 )
THEOREM U . 2 . 5 . - Let {M > {M*
T!
and
(U.2.3).
be sets of c.inde.modules such that
{M^lj is a semi-T-nilpotent system with respect to J
!
and £ ©M^
K
satisfies the property III for any direct summand of it. Then
M» Z$M «l0Mg satisfies the property III for any direct summand of M.
J
K
a
69
Applications of Factor Categories • • .
Proof. - Let M = H^Ng. Since £ ©M^ = N
J
q
has the exchange property in M
by (1+.1.3), M = M^N^CNg', vhere Nj = Hj'Wff.''. Hence, ^ / m * N ' O T g ' ^ E f M ^
K
q
1
!,
M
Therefore, N.'G A from the assumption. On the other hand, N^ iN2 ^M
o
and
hence, N/'c A by (3.1.U).
COROLLARY U . 2 . 6 . - Let M = I W
^.
and M
Ot
c.inde. .Let {M }„
Ot
Q
be the subset
p J\
of {M } which consists of all countably generated R-modules. If
!
{M^}^_£ is a locally semi-T-nilpotent system with respect to <7 ,
then the property III is true for M.
Proof. -
It is clear form {k.2.k) and ( U . 2 . 5 ) .
Finally, we add here a corollary to (U.2.U).
Corollary U . 2 . 7 . - Let M,
be as in (4.1.3). If ft is ^-projective,
A. Especially if M is ^-projective, the property III is true
3
for M.
Proof. -
Every R-projective module is a directsum of countab]^ genera­
ted R-modules by (U.2.3) and hence, N^'A by (U.2.U).
70
Applications of Factor Categories • . .
CHAPTER 5 . SEMI-PERFECT MODULES
H. Bass [ 2 ^ defined semi-perfect or perfect rings as a generalization
of semi-primary rings in i 9 6 0 . Later E. Mares [28] succeeded to generalize
them to modules in 1963 In this chapter we shall give many interesting properties of semiperfect modules given by [ 1 9 , 28] . We always assume that a ring R
contains the identity and modules are right R-modules and unitary.
5 . 1 . Semi-perfect modules
Let M ?N be R-modules. If
any submodule T of M with property :
M = T+N always coincides with M, N is called small in M.
;
LEMMA 5 . 1 . 1 . - Let Ac B £ M C N be R-modules. Then
1) If B is small in M, then A is small in N.
n
2) Let ( A ^ ^
be a finite set of small submodules in M , then
is also small in M.
!
3) Let f be a homomorphism of M to M . If A is small in M,
f (A) is small in M*.
Proof. - It is clear from the definition.
DEFINITION. - Let
M — ^ 0 be an exact sequence of R-modules. If P
is R-projective and KerTT is samll in P, we say P is a projective cover
of M. We shall denote it by (P,7l) and P by P(M), respectively.
71
Applications of Factor Categories .
LEMMA 5 . 1 * 2 . - Projective covers (P,7l) of M are unique up to isomorphism
if they exist. If
P — ^ M — ^ 0 is an exact sequence with P
1
projective then (P,7r) is naturally imbedded in P
3
1
1
as a direct
swmand.
Proof. - From a diagram ;
p
r
>M
*0
• \ i <
we have 0 and P = Im 9+Ker 7f , since P
f
is projective and f is surjective.
f
Hence, P = Im0, which implies P =Po®Ker6, since P is projective. The first
part is clear from the last.
DEFINITION. - Let P be an R-module. If P is R-projective and every fac­
tor modules of P have projective covers, we call P semi-perfect. If
every direct sum of copies of P are semi-perfect, we call P perfect.
LEMMA 5 . 1 . 3 [ 2 , 2 8 ] . - Let M be
Let 0 ;M
u
semi-perfect and U a submodule of M.
M/U be the natural epimorphism. Then there exist pro­
jective submodules P and V of M and of U, respectively such that
M = P 9 V, i£ |p —^M/U ts a projective cover and UnP is small in
P.
Proof. - Take a diagram ;
P(M/U)
* M/U
72
0
Applications of Factor Categories . . .
Then M = P $ Ker f "by ( 5 . 1 . 2 ) , where P
<4
P(M/U) and P^ U is small in
P. It is clear Ker f e U .
COROLLARY 5 . 1 . * + . -
Let M be semi-pefect.
M, U is small in M or there exists
such that
M
Proof.
-
Then for any submodule U of
a non-zero
direct
svomand V of
U?V.
If U is not small in M, U?Ur>P
by ( 5 . 1 . 1 )
and ( 5 - 1 . 3 ) . Hence,
P^>U and so V / 0 .
LEMMA 5 . 1 . 5
Then
[373*- Let P be R-projective
and S = End(P).
p
J(S) = [f jes, Im f is small in PJ .
Proof. - Denote the set of right side in ( 5 - 1 . 6 ) by J'(S). It is clear
from ( 5 . 1 . 1 ) that J (S) is a two-sided ideal in S. For any f £ S we have
f
P = Im f + Im ( 1 - f ) . Hence, if f CJ'(S), P = Im ( 1 - f ) . Since P is pro­
jective, P = Ker (l-f)«P . Put K = Ker ( 1 - f ) . Then K=f(K) cf(P), which
f
is small in P. Hence, P = P and K - 0 . Therefore, J (S) cj(S). Conver­
f
f
sely, let g<rJ(s). We shall show that g(P) is small in P. Let P = T+g(P)
for some T C P and consider a diagramm ;
P
L>
P
^
P/T
( J g is surjective)
%
k
P
Then
(1-gk)=0 and hence, ^ - 0 , since gk€J(S). Therefore, P = T.
73
Applications of Factor Categories . • •
PROPOSITION 5 . 1 . 6 . -
Let M be a semi-perfect module. Then S/J(S) is a
regular ring in the sense of Von Neumann, where S = End (M),
R
(cf. [23, 28]).
Proof. - Let s G S . Then there exists a submodule P of M such that
M = Im s + P and Pn Im s is small in M by ( 5 . 1 . 3 ) .
morphism
We define an R-homo-
1
1
$ : M/P — ? M/s" (P) by setting <J>(s(m)+P) = m+s~ (P), which
is clearly well defined. Now consider a diagram ;
M
^ 0
1
>M/s" (P)
T
(J>
M/P
/>
x ^
\
j
P
* M
1
Then ts(m)-me s (P) and hence s(ts(m)-m) € P r\ Im s. Therefore,
s-sts6J»(S) = J(S) by ( 5 . 1 . 5 ) .
For any R-module A we put J(A) = n (Maximal submodules in A) or
J(A) = A is there exist non maximal submodules. If A = R, J(R) is the
Jacobson radical of R. We note that every small submodule in A is contai­
ned in J(A) and that f(J(A)) CJ(B) for any R-homomorphism f of A to B.
From now on, we shall denote Hom (A,B) by [a b\ and EndpCA) by S^.
R
9
PROPOSITION 5 . 1 . 7 . - Let P be R-projective. Then J(P) is small in P if and
only if j(Sp) = [P,J(P)] . In this case S /J(S )^End (M/j(p))
p
rings.
74
p
R
a
8
Applications of Factor Categories . . .
Proof. - From the above remark we always have J(Sp) c[p j(pj] by ( 5 . 1 . 6 )
f
for projective P. If
J ( P ) is small in P, [P,J(P)] ^ J ( S
p
) by ( 5 - 1 . 5 ) .
Conversely, suppose [P,J(P)] = J(S ) and P = J(P)+N for some submodule
p
N. Then we consider a diagram {
J(P)
> J ( P ) / K C\ J ( P )
* 0
8
v
P/N
p
From it we obtain J(P) = h(P)+NoJ(P) and hence, P = N+J(P) = N+h(P).
Since
h€[P,J(P)] = J(S ) , h(P) is small in P. Therefore, P = N and we
have shown that J(P) is small in P. Since P is projective, we have an
exact sequence ; 0 -* [ P , J ( P ) J
S -^ [p,P/J(P)J — 0 . It is clear
that [P,P/J(P)]
by the above remark.
p
= [P/J(P),B/J(P)J
LEMMA 5 . 1 . 8 . - Let {A }_ be a set of R-modules such that [ A ,J(A )]sJ(S )
ot l
a
ot
A
J
for all a el. Put A = I © A
I
then Im f 4 J(Im f).
. If Ker (1-f) 4 0 for some f € S
a
A J
A
Proof. - Put B = Im f and suppose B = J(B). Since J(B)cj(A), f£(A,J(A)]«
Ker (1-f) 1 0 implies that there exists a subset {l,2,...,n} such that
n
( Yl ® A.)oKer (1-f) 4 0 . The following argument is analogous to the
i=1
1
proof of ( 2 . 1 . 1 . ) . Let e be the projective of A to A^. Since
1
f€[A,J(A)] , e fe JA €[A ,J(A )] £ J(S ) by the assumption. Hence,
1
1
1
1
1
A
75
Applications of Factor Categories . •.
(l-f)e
(1-f )e j A^ is an automorphism of A
4|
1
so A = (l-f)(A )©Ker e, = (l-f)(Aj • I
1
:
e
1
1
A — A
1-f
$ A
and A, ^
1
and
(l-f)(Aj
Now, we repeat the same argument on the latest decomposition and on A^.
Then we have A = (1-f) (A )©( 1-f) (A ) • H
©A . Finally, we have that
a=1,2
n
(l-f)J(H ©A.) is isomorphic from this argument, which is a contradiction.
1
1
Hence, B # J(B).
COROLLARY 5 - 1 . 9 f 2~\. - If P is ^-projective, P 4 J(P) and J(P) = PJ(R).
Proof. - It is clear J(R) = £ r , J ( R ) J and P is a direct summand of copies
of R. Hence, P^J(P) from ( 5 . 1 . 8 ) . The last part is also clear.
We note that ( 5 - 1 . 9 ) shows that P contains a maximal submodule.
COROLLARY 5 - 1 . 1 0 . - If M is semi-perfect, J(M) is small in M.
Proof. - 1. y ( 5 - 1 - U ) either J(M) is small in M or J(M) contains a non­
zero submodule 7 such that M = V © V . If we had the latter, then J(M) =
f
J(V)$J(V) and J(V) = J(M)AV = V. Hence, V = 0 by ( 5 . 1 - 9 ) .
PROPOSITION 5 - 1 - 1 1 -
-
Let M be semi-perfect. Then M/J(M) is a semi-
simple module.
Proof. - M M *
M/J(M)
and U = U/J(M) for a submodule U ? J ( M ) .
By ( 5 - 1 . 3 ) there exist submodules P,V in M such that M = P©V, V c U and
UfiP is small in M. Then W P c J(M) . On the other hand, (P+J(M))o U =
(mu)+J(M) = J ( M ) . Hence, M =
P9U. Therefore, M is semi-simple (since
76
Applications of Factor Categories . . .
R contains the identity or J(M) ^ M ) .
LEMMA 5 - 1 . 1 2 . - Let P be an R-projective module such, that J(P) is small
in P. Suppose that P/J(P) is a direct sum of submodules { P ^ j &s
R/J(R)-modules and that for each ael, there exists a projective
module Qa/JlQ^^ P^ • Then the above decomposition of P is lifted
to P.
Proof. - Put Q = £ « Q
I
. Since P
P/J(P) is a projective cover of
a
P/J(P), P/J(P)*- Q/J(Q) and Q is projective, Q = P©Q' by ( 5 - 1 . 2 )
0
»J(P)
f. P
» P/J(P)
= I
>r
6
V
f
A
I
:
J P ' - ^ O
a
'
Q/J(Q)
-
Z
* Q.
1
f,
Q
Then Q = P+J(Q) = P*J(Q') and hence, Q' = 0.
COROLLARY 5 . 1 . 1 3 . - Let Vibe semi-perfect and M/J(M) = I ©M^ .
Then there exists a decomposition of M
M = 2 QM which induces the
I
above. Especially M
is a
a
direct sum of c.inde. modules.
Proof. - We know from the proof of ( 5 . 1 . 1 1 ) that M satisfies the condi­
tion in ( 5 . 1 . 1 2 ) . Hence, we obtain the first part from ( 5 . 1 . 1 2 ) . Since
M/J(M) is semi-simple by ( 5 . 1 . 1 1 ) » M = T. «M"
j
P
where M" /J(M") are miniB
P
mal by the first part. Since End(Mg/J(Mg)) = End(Mg)/J(EndMg) by ( 5 - 1 . 7 ) ,
Mg
is c.inde..
77
Applications of Factor Categories • • •
From this corollary we can apply the results in the previous
chapters to semi-perfect modules.
THEOREM 5 . 1 . U [28}--
Let M be semi-perfect. Then we obtain
1) J(M) is small in M.
2) M/J(M) is semi-simple.
3) Every decomposition of M/J(M) such as M/J(M) = M^GMg
1
ts
lifted to M.
Conversely, if a projective module M satisfies 1) - N _ 3 ^ t/ien M is
s^mt-per/get.
Proof. - We have shown the first half. We assume a projective module
M satisfies 1 ) ~ 3 ) . Let A be a submodule of M and put M = M/J(M) and A
(A+J(M))/J(M). From 2 ) and 3) there exist submodules M ,Mg such that
1
M = M^Mg and M
= A. Then we have a diagram ;
0
M/A —-t
- M/A
Y
0
X
M
f \
2
\
I"
\
\
e
\
Ker <J = (A+J(M;;/A is small in M/A by 1 ) and (5.1.1). Hence, f is sur­
jective. On the other hand, Ker f cKer e = J(M ), which is small in M
2
by 1 ) . Therefore, (M ,f) = P(M/A).
78
g
Applications of Factor Categories . . .
5 . 2 - SEMI-T-NILPOTENCY AM) SEMI-PERFECTION
We have shown by ( 5 - 1 . 3 ) that every semi-perfect modules are
directsums of c.inde.projective modules. In this section, we shall consi­
der the converse case.
THEOREM 5 . 2 . 1 . - Let { }j be a set of projective modules P and
p
a
a
p
P = Z $P . Then J(P) is small in P if and only if J ( ) is small in
a
P
a
for all acl and {P } is a locally semi-T-nilpotent system
T
p
with respect to the radical J (of the induced category from t ) j ) .
a
Proof. - If J(P) is small in P then J("P ) is small in F„ by ( 5 - 1 . 1 ) .
*
Let
V
a
ea
{P.}~
*>
I i
=
ot
subset of {P a} I and {f.1 : P.1
P.1 +, 1and f.frj}.
Put
i
T
+f
^Pi i(Pi) U
P; ® P1
is small in P ^ P ^ ,
J
<
p
•P.PiCPjh Since J(P.) ® (
)
CO
f (p ) € j ( P ) by ( 5 . 1 . 7 ) . Then P = XL +P ' +
+1
1 +
i
i + 1
i
i+1
i
oO
Z
+P+J(P). Since J(P) is small in P, P = ]T «P.'€ £
Y*U)
i-1
Y*(i)
Y
1
1
Hence,
y
(Py}^. is a locally semi-T-nilpotent system from {*** ) in the proof of
( 3 . 1 . 1 ) . Conversely, we assume that J(P ) is small in P^ for all a €1 and
a
{P } is locally semi-T-nilpotent. Then [ P .J(P )l = J(S ) by ( 5 . 1 . 7 ) .
Ot x
* Ot
Ot *
jt
rv
p
a
p
J
p
We shall put C n f p , p ] « [ > ( g)l
a
conditions
i n
a
( 2 - 2 - 3 ) . Then C satisfies all
in ( 2 . 2 . 3 ) . Hence, [P,J(P)j c J(S ), which implies that J(P)
p
is small in P by ( 5 . 1 . 7 ) .
79
Applications of Factor Categories • . .
COROLLARY 5 . 2 . 2 . - Let { ^j and
p
a
if only if P^
p
be as above. Then P is (semi-)perfect
s
is (semi-) perfeet and {^^j ^
a
locally (semi-)T-
nilpotent system with respect to J.
Proof. - We assume that P is semi-perfect• Then each P
and J(P) is small in P by ( 5 . 1 . 1 U ) . Hence,
is semi-perfect
a
s
^ locally semi-T-nil-
pofcent. If P is perfect, consider any co-products of copies of P, then
the above argument shows that {P^}^ is locally T-nilpotent. Conversely,
we assume that each P
i s s e m i - perfect.
Then by ( 5 . 1 . 1 1 ) and ( 5 - 1 . 1 3 )
f
!
P/J(P) is a semi-simple module and P = £ ®^ g > where P g are c.inde..
J"
Since {
p
a
} j is
a
locally semi-T-nilpotent system with respect to J, so
is {P }j • Furthermore, J n[P£l,P£] = J £ g> gl » (
f
f
n
p
p
e
see
§
f
o
r
the
1
definition of J ) . Hence, every idempotent in Sp/0"(Sp) is lifted to Sp
by ( 3 . 2 . 5 ) . J(P) is small in P by ( 5 . 2 . 1 ) . Therefore, P is semi-perfect
by ( 5 - 1 - 1 * 0 . If ^ ^ j is locally T-nilpotent, we can use the above
p
a
argument on any co-products of copies of P. Hence, P is perfect.
COROLLARY 5 . 2 . 3 [ 3 3 , 3 6 ] . - Let S be any ring with radical J(S) and (S)^.
the ring of column finite matrices over S with any degree
I.
Then J((S ))= (J(S)) if and only if J(S) is right 1-nilpotent.
I
Proof. - Put M = [ I S , then [M,M] = ( S ) and [M,J(M)] * ( J ( S ) ) . Hence,
I
T
T
(J(S)) « J((S) .) if and only if J(M) is small in M by ( 5 . 1 . 7 ) and hence
I
]
if and only if J(S) is right T-nilpotent by ( 5 . 2 . 1 ) .
80
Applications of Factor Categories . . •
THEOREM 5-2.U. - Let P
be an indecomposable and projective modules.
Then P is semi-perfect if and only if P is c.inde. .
Proof. -
If P is semi-perfect, P is c.inde. by ( 5 - 1 . 1 3 ) - The converse is
a special case of the following theorem.
THEOREM 5-2.U'. - Let P be projective, then we haue the following
equivalent statements.
1) Sp
is a local ring.
2) Every proper submodule of P is small in P.
3) ? is semi-perfect and indecomposable.
Proof. - l ) - * 2 )
Since S
p
is local, P is c.inde. and hence, J ( S ) consist
p
of all non-isomorphisms in Sp. Let N be a proper submodule of P and
P = T+N for some submodule T in P. Then we have a diagram ;
0
— » T^N — * N — *
\
N/N o T
>> 0
*
P
Since N i P, a€j(S ) and N= TON+Im a. Hence, P = T+Im a. Since Im a
p
is small in P by
2)
(5.1-5),
P=T.
1 ) Let f i 0 € S
p
be a non-isomorphism. If Im f = P, P * P «Ker f,
Q
since P is projective. Hence, Ker f = 0 by 2 ) , which contradicts
the
assumption. Therefore, Im f 4 P- Let g be another non-isomorphism in Sp.
81
Applications of Factor Categories . . .
Then P^Im f + Im g?Im(f+g). Hence, the set of non-isomorphisms in S
is the two-sided ideal, which means that S
p
p
is local.
2)
3) It is clear.
3)
2 ) Let T be a proper submodule of P and P = P(P/T). Since P is
1
projective, P = P'GP" by ( 5 - 1 . 2 ) . Hence, P = P and T is small in P.
f
REMARK. - If P is semi-perfect and indecomposable, J(P) is a unique
maximal submodule of P by ( 5 . 2 . U ) , 2 ) . Hence, P^eR for some idempotent
1
e, since P is cyclic. Thus, there exist semi-perfect modules if and only
if R contains a local idempotent e, i.e. eRe is a local ring.
COROLLARY 5 . 2 . 5 . - Let P be a semi-perfect. Then there exist maximal ones
among perfect direct summands of P and those modules are isomorphic
each other.
Proof. {P )
y o
Let P = I" eP^
of {P„K such that
T
Ot 1
and P
c.inde.. Let £ be the set of subset
a
{P,,} is locally T-nilpotent. We can find a
y <J
T
maximal one in S by Zorn's lamma, say (P.}
since {P } is semi-T-nil-
Tt
T
potent. Put P = Z ®Pyi then P is a desired perfect summand of P by
(5.2.3).
Let P = Z e P G l G P . = I « P « T « P ' , where £ • p
J
K
J'
K
J
Q
q
1
T
0
Y
f
0
Y
and Z $ P• are maximal perfect submodules. Then P and P' are themselves
j,
Y
T
T
T-nilpotent, respectively. Hence, if P^ is isomorphic to some P ^ in
M
{PL
o iv
7
) ,P} is locally T-nilpotent. Which contradicts to the
y J y
T I
maximality of £ 9 P' ' . Therefore, Z 9 P
J
J
^
1
82
^
«P' , by ( 2 . 1 . 1 + ) .
Applications of Factor Categories . . .
PROPOSITION 5 . 2 . 6 , - Let P be semi-perfect and P
in P. Then P
Proof. -
Q
Q
a projective sub-module
is a direct summand of P if and only if J(P) n ?
Suppose J ( P ) = J ( P ) q P
o
there exists a direct summand P
- J(P )
.then P /J(P )cp/j(P). By ( 5 - 1 . 1 3 )
o
o
O '
of P such that P
1
Q
/J{P^)eP /J(P )=P/J(P).
Q
0
On the other hand, the formal directsum P ^ P is isomorphic to P by
Q
( 5 . 1 . 1 2 ) . Hence, J(P ) is small in P . Consider a diagram ;
Q
0
> J(P )—> P
0
*
P/(P +J(P))
1
£
f
> 0 (exact)
1
\
\
g
\
\
N
P
where i is the inclusion. Then (l -gi)(P )C J(P ) and hence,
o
(l -gi)€ J(S ) by ( 5 . 1 . 5 ) . Therefore, gi is isomorphic in S
p
p
Q
Q
p
0
, which
p
o
0
means that P is direct summand of P. The converse is clear.
o
5 . 3 . PROJECTIVE ARTINIAN MODULES.
Let M be an R-module. If for every series
submodules M. of M there exists n such that M
1
M
artinian. Let T be'a subset of
n
I^J?
= M
... ?M^ ? ... of
. for all t, we call
n+t
We put TM = {f(m)|f C T and m<sM}.
p
LEMMA 5 . 3 . 1 . - Let M be artinian and projective• If AM = A
M ^ 0 for a
right ideal A in S^ , Then A contains a non-zero idempotent.
Proof. - Since M is artinian, there exists a minimal submodule N = A'M
with respect to properties N
!
= A"M = A M } 0 for a right ideal A"c A.
I?2
83
Q
Applications of Factor Categories . . .
Then A
1
is not nilpotent. Hence, there exists x in A
1
!
such that xA 4 0.
Again from the assumption we can find a minimal one among submodules
x M,(x €A) and x M i 0 , say xM, (xeA). Since xA'A'M = xA M 4 0 , there
f
f
!
!
exists yG xA' such that yA' 4 0 . Then yMcxA'McxM. Hence, yM = xM by
the minimality of xM. Now, consider a diagram ;
^—* yM = x M — > 0
M
\
Y
^
x
\
M
2
Then x = yr = xa, where a£A • Hence, x = xa = xa
2
= ... . Therefore,
2
a is not nilpotent and x(a-a ) = 0 . Put n = a -a. If n = 0 , a is a non­
zero idempotent. Suppose n 4 0 . Put A* - | zJ£A , xz = 0J , then
f
1
A o A*3 n. We consider a series ; A* M p A ^ ^ p .. . ofi^Vl ? ... . Since
4n
M is artinian, k M
= A*
n + 1
for some n. Since A'MjpA^M and A'M is the
minimal one, A'M = A M or A* ^ = 0 . On the other hand, xA 4 0 and
#n
1
1
•#
n
•
xA = 0 and hence, A* = 0 , which implies that n is nilpotent. Next, put
a^ = a+n-2an, then all a,n and a^ commute each other, since they are
generated by a. Hence, (-n+2an) is also nilpotent and a^ is not nilpotent.
2
2
Furthermore, a^ -a^ = n (Un-3). Repeating this argument we get nonnilpotent elements a. £ A
f
such that (a.-a.^) = n^z., z. € S . Since n is
W
l
1
1
1
1
M
1
nilpotent, we have a non-zero idempotent a^ in A .
COROLLARY 5 - 3 . 2 . - Let M be as above. Then S^ is a semi-primary ring.
84
Applications of Factor Categories . • .
Proof. -
Since M is artinian, M is a finite directsum of indecomposable,
projective module NL. First we assume M = M^, For any right ideal A in
S„A
=A
n +
M for some n. If A
4 0, A contains a non-zero idempotent
e by ( 5 . 3 . 1 ) . Since M is indecomposable, e = 1 . Therefore,
is a local
n si
ring with nilpotent radical. Next, we may assume M = £ £ © M. . 9 where
i=1j=1
J
M..S are indecomposable and M. . ^ M - . , , M . . < £ M . .
\i
- sj
J
.
'
F
1
1 J
1 J
1
!
, if i ^ i . Then
J
s
M •
=
£ ij)l ij iji J
{8
S
8
Gs
r
V
L *
/*— 1
' * ik-ll•
M
s i n c eM
i sc > i n d e
io
-
jv — 1
from the above,
J
/
(
s
S
1 1
}
12
S
J(S J
Q 1
9
2 1
J(s ) =
S
2
in
\
...
\
2
2
n
M
\ S ,
J(S ) /
nn /
n1
by ( 2 . 1 . 3 ) . Furthermore. All J(S..) are nilpotent and hence, J(S ) is
n
nilpotent and S /J(S )
• S ^ A K S ^ ) . It is clear that S^/JlS^)
i=1
M
M
M
1
1
is the ring of matrices wer a division ring S^ ^^y[ )•
i1
i1
LEMMA 5 . 3 . 3 . - Let M be R-projeotive and A a finitely generated right
ideal in S^. Then A = £ M , A M ] . Furthermore, if M is R-finitely
generated, A' = ["M,A M] for any right ideal A
!
85
1
in S^.
Applications of Factor Categories . . .
Proof. -
n
Let A = XT
a
i ^ . Then we shall consider a diagram ;
i=1
I
AM
©M.
>0
\ M
where M ^ M for all i, 4>=(a ,a^... ,&^) and x is any element in £~M,AM] .
1
b
I
: \
T h e n
We shall denote h by
2 |•
[^MjAMJS A. It is clear
A<J{M,AM3
n
XT
i=1
• M.
1
by
H 9 M
a^A
x
=
. If
H
M
•
a
h
i i
l s
l n
•
A #
Hence,
is finitely generated, we replace
t
in the above, then h(M)C H
a
i=1
9 M
i
. Hence, we
a
can make use of the same argument.
THEOREM 5.3.1*. - Let M be ^-projective and artinian. Then M is a perfect
R-finitely generated module and
is right artinian.
n
Proof. - It is clear from the proof cf (5.3.2) that M = £T © M., where
i=1
1
s are cinde.. Furthermore, since
1
is semi-primary by (5.3.2), M^
is a (locally) T-nilpotent system with respect to J. Therefore, M is
perfect by (5.2.2) and (5.2.U) and M^ is cyclic. Furthermore, (5.3.3)
gives a lattice monomorphism of the set of right ideals in
set of submodules of M. Hence,
is right artinian.
86
into- the
Applications of Factor Categories . . .
CHAPTER
6.
INJECTIVE
MODULES
In this chapter we assume that the reader knows elementary proper­
ties of infective modules and we refer to f 8 } for them.
We mainly study some application of ( 1 . 3 . 2 ) to injective modules
and hence, we shall consider directsums of indecomposable and injective
modules. We reproduce [ 1 0 , 2 5 , 2 9 , 31 , ^o] by virtue of factor categories
and study the Matlis'problem in § 6 . 5 .
6 . 1 . ENDOMORPHISM RINGS OF INJECTIVE MODULES.
In this section we shall recall some properties of the endomorphism
rings of injective modules, which we make use of later. If the reader is
not familiar to them, consult [ 8 ] .
11
As a dual of the concept "small , we shall define the concept
"large". Let M ? N be R-modules. If for any son-zero submodule T of M,
N O T # 0 , we say N is large submodule in M or M is an essential extension
of N. We denote it by M O N .
As a dual of ( 5 . 1 . 6 ) we have
LEMMA 6 . 1 , 1 . Let E be infective and S
E
= Ehd(E). Then J(Sg)
s
{f|€S , Ker fcE} and S^/JiS^) is a regular ring.
E
As a dual of
(5.1.1U).
LEMMA 6 . 1 . 2 . Let E and S_ be as above. Then a finite set of mutually
orthogonal idempotents in S /J(S ) is lifted to Sj.
E
87
E
Applications of Factor Categories • • .
As a dual of projective cover, we define an injective envelope
(injective hull) E of R-modules M as follows ; E is injective and M is
large in E. Contrary
to projective covers , every modules have injective
hulls and every injective hulls are isomorphic (dual to ( 5 . 1 . 2 ) ) . Hence
by E(M) we shall denote an injective hull of M.
6 . 2 . CATEGORIES OF INJECTIVE MODULES.
We shall give here an application of ( 1 . 3 . 2 ) to injective modules.
Let M be an R-injective module. We shall define a full sub-additive
category £(M) in M^ as follows (cf. the induced category in § 1.U) ;
the objects in £(M) consist of all direct summands of any products
7T
If M is an injective and cogenerator in M
is
^ M ot ; M^^M.
ot
—-n , then C(M)
—
D
the category of all injective modules. We also call C^M) the category of
injective modules induced from M. Let J be the radical of C(M) (see § 1 . 1 .
for the definition).
THEOREM 6 . 2 . 1
[ 1 7 , 39] . - Let M be and R-injective module and C(M) the
category of injective modules induced from M and J the radical of
£(M). Then C(M)/J is a Grothendieck and spectral category.
Proof. -
We shall denote C(M)/J by C(M). Then 5(M) has a finite coproducts
from the definition and Remark 2 in § 1 . 1 , and £(M) is a regular category
from ( 6 . 1 . 1 ) . Furthermore,
C(M)
( 6 . 1 . 2 ) shows that £(M) is amenable. Hence,
is an abelian spectral category by ( 1 . 3 . 2 ) . We shall show that C(M)
88
Applications of.Factor Categories • . .
has finite co-products. Let
A
be a set of objects in C(M). Since
< • K M, I 9 k
X
1
a
and E = E( £ « A ) is an object in C(M),
a
J
1
a
since TC TM is injective. We show E = If $ A^ . Let N be any object in £(M)
and {f :A^ + N} a set of morphisms, where f
Then there exists f : £" $ A^ + N in
A
•+ N is a representative.
such that
*I $ A
a
:A
*E
a
f
« \ i '
N
Since N is injective, there exists g : E
N which commutes the above
diagram. We can easily show from ( 6 . 1 . 1 ) that g does not depend on a
choice of representative f
proof of
(3.2.7)).
and that g is uniquely determined,(cf. the
Also we can similarly show that for a given g:E
there exists a unique set of f
E =
:A
N,
N
such that "g = irf
Hence,
a
a
°
a
A . Next we shall show that C(M) has a generator. Let S be the
set of right ideals K in R such that E-, = E(R/K)&C(M). Put U = Z $\.
~~
K€S
K
K
Let T be an object in C(M) and t ^ OeT. Then TjtR-^R/(0:t) and
E
E(R/(0;t) )€ ¡3, since T is an injective and in C(M). Therefore, ( -t)
r
0
is isomorphic to a direct summand of T, which implies [U,TJ ± 0. Finally,
we shall show similarly to the proof of (1.U.8) that ((J A )s\ B = vj(A n B)
K
K
89
Applications of Factor Categories . . .
for a subobject B and a directed set of subobjects {A^}^ in a given object
F. Put C = U (1 /IB), then B = C©S and ( U A ) nB = C 0 (( U A )
a
o
a
K
a
K
rs\).
o
K
We put D - ( U A )^~B and assume D ? 0 . From an exact sequence
K
°
a
T ©A
K
fg =
v e
U A —>°9
8
obtain a monomorphism g : T ) - 9 Z * A
8
such that
a
K
i
We note that g is R-monomorphic, since J is the Jacobson radical
a
3>
and that 2 • £
= E(2" 6 A ). Put D' = Im g in JL. Then D
f
= Im g. Since
f
4 0 in JL, Let x ± 0 be an element in D'O.?* A,, and
K
n
let E(xR), E-fxR) be injective hulls of xR in D and Z" $ A 2 , respectii=1
n
n _
vely, where x€
9 A
. Then E(xR) = E^xRjc £ e A
from Remark 2
i-1
i
1
i
below. Hence, E(g" (x)R)c UA C A for some 6 such that 3 > a. and
D
f
1 0 , D /0 Z $ A
K
1
a
a
1
0
cu~
1
p
1
E(g (x)R)c D, which is a contradiction.
REMARKS 1 . We noted in the proof of (1.U.8) that Z © M = Z 9 M
I
I
factor category of cinde.modules. However, in £(M) 71 9 A
I
in the
is not, in
a
general, an object in £(M) and £ 8 A means E( Z • A ).
I
I
2 . Let E, E be injective and f : E — ^ E . We shall find Ker f
a
1
1
and Im f in C(M). Let K = Ker f in
We define f'€ [e,E'3 by setting f
f
and E " - E(K) in E. Then E * E «E .
II
1
= (0,f I E ^ . Then Ker (f-f• ) « k § e C E.
Hence, f = f . Therefore, Ker f = Ker f = E" and Im f = Im f = f(E ) .
1
1
1
This argument shows that Ker f (Im f) does not depend on a choice of
injective hulls of K in E and that we can give direct proofs of many
90
Applications of Factor Categories . • •
results in the following without factor category. However, if we use
the factor category, the proofs are simple and natural in some sense.
3. If J # 0, JTA 4 ^ A
a
for A e C(M) in general.
a
h. Instead of injective modules, we can consider the full sub­
additive category P of projective modules in M^. However, in this case
P/J is not spectral. We know that P/J is spectral and Grothendieck category
if and only if R is right perfect ring (see [ 1 9 ] ) .
For any R-module M we put Z(M) = £ m | 6 M , (0:m) cR] . It is clear
r
that Z(M) is an R-submodule of M and we call Z(M) the
singular submodule
of M.
LEMMA 6.2.2. - Let M be an injective module with Z(M) = 0, then J(S ) = 0.
M
Proof. -
Let f€J(S ). Then Ker f cM and so Z(M/Ker f) = M/Ker f. On the
M
other hand, M/Ker f is isomorphic to a submodule of M. Hence, M = Ker f.
PROPOSITION 6.2.3. - Let M be an injective R-module with Z(M) = 0. Then
£(M) is a spectral and Grothendieck category with generator M.
For any morphism f in £(M), Ker f(lm f) in £(M) is equal to
Ker f
Proof. -
(Im f) in-Mp.
From (6.2.2) we obtain J = 0. Hence, £(M) is a spectral and
Grothendieck category. Furthermore, since M is a cogenerator in C(M), M
is a generator. The remaining part is clear from Remark 2.
91
Applications of Factor Categories .
COROLLARY 6 . 2 . U . - Let N be an R-module with Z(N) = 0 and Q-j >Q infective
2
an
submodules in N. T/z^n Q^Qg ^
^
injective.
Proof. - Let E = E(N) and consider £(E). Then Q e C(E) and Q^Qg nd
a
Q
¿
are an image and a kernel in
, they are injective in
2
of morphisms in £(E), respectively. Hence
by ( 6 . 2 . 3 ) .
LEMMA 6 . 2 . 5 . - Let B be a full sub-additive category in M^. Suppose B
contains a generator (cogenerator) in M^. Then every monomorphism
(epimorphism) in B
is monomorphic (epimorphic) in Mp.
Proof. - Let U a generator in M^ , which is contained in B and f:A-=^B
a monomorphism in B. Put Ker f = C in L . If C
g i- 0€.[u,cj
9^0, there exists
in Mp such that ig i- 0 , where i:C -*>A is the inclusion.
However, igé[u,A] eB and fig = 0 , which is a contradiction.
PROPOSITION 6 . 2 . 6 . - Let M be an R-injective module. We assume M is a
generator and cogenerator in M^, (e.g. R t s a Q . F . ring). Then
C(M) is an abelian category if and only if R is a semi-simple
artinian ring.
Proof. - We assume £(M) is abelian. We shall show for any morphism f in
f
i
C(M) that (Ker f in C(M)) = (Ker f in M^). Let f : N
decomposition of f in £(M). Since £(M) is abelian, f
Im f
1
N
1
be a
is epimorphic in C(M)
and i is monomorphic in £(M). Hence, so are they in M^ by ( 6 . 2 . 5 ) . Hence,
(Im f in £(M)) = (Im f in M^). Put K
1
= (Ker f in C(M)) and K
92
g
= (Ker f in
Applications of Factor Categories . . •
by ( 6 . 2 . 5 ) . On the other hand, K is R-injective and
f .
• N" in M^. Then N €: C(M) and N" < = ^ ^ (In f in Mp)
It is clear L C L
hence, N = K
ff
1
from the above. Hence,
n
= Kg. Let A be any R-module, then there exists
an R-exact sequence ; 0 -> A — ) T M
17 M. Since ?TMe£(M), A = (Ker f
in M^) = (Ker f in £(M)). Hence, A is injective. Therefore, R is semisimple and artinian. The converse is clear.
6 . 3 . DECOMPOSITIONS OF INJECTIVE MODULES.
This section is a reproduction of [ 2 9 ] by virtue of factor cate­
gory and we shall give a condition under which every injective module is
an injective hull of some direct sum of c.inde. modules, which is equi­
valent to a fact that A/J is completely reducible, where A is the full
sub-additive category of all injective modules in M^.
LEMMA 6.3.1.<*£et & be a full sub-additive category in M^. We assume that
every direct summand in M^ of an object in B belongs to B. Then
every finite co-product in B/J is lifted to Mp.
Proof. -
Let B, B
1
and Bg be in B and B = B^Bg in B/J. Then there exist
morphisms i^rB^^ B and P".B
k
p
k^k
=
1
S
Bk* ^
n c e
B^ such that 1
+
g
P
= ^-|Pi ^2 2
a n d
J is the radical, P^i^ is isomorphic in M^. Hence,
M = Im i^Ker p
in M^. By the assumption Im i and Ker P ^ B
1
1
-
clear that Ker p
1
= Bg and B = Im
$ Ker p^ = l^SBg.
93
and it is
Applications of Factor Categories . . .
COROLLARY 6 . 3 . 2 . - Let M be R-injective. Then an object N in £(M)/J is
minimal if and only if N is indecomposable.
Proof. - It is clear from ( 6 . 2 . 1 ) and ( 6 . 3 . 1 ) .
PROPOSITION 6 . 3 . 3 . - Let R be a left perfect ring and M R-injective as a
right R-module. Then £(M)/J is a completely reducible and Grothendieck category.
Proof. - Since R is left perfect, every right R-module contains minimal
submodules by ( 2 ] . Let N be in C(M) and S(N) the socle of N in M^, i.e.
S(N) = 2 ®_J
an<
a
* ^ '
a
s a r e
minimal R-modules. We know from the assumption
that N P £ « I . Hence, N = Z «E(I ) by Remark 1 and E(l ) is a minimal
ot
object in C(M) by
ot
ot
(6.3.2).
Let A be the full sub-additive category of all injective modules
in Mp. By A we shall always denote A/J in the follows. We know from
( 6 . 3 . 3 ) that if R is a left perfect ring, then A is completely reducible.
We shall give a condition for A to be completely reducible £ 2 9 j .
DEFINITION.-Let K be a right ideal in R. K is called reducible if there
exist right ideal K
i
in R such that K =
K
g
and
4 K. If K is not
reducible, we call K irreducible.
We shall denote E(R/K) by E .
v
LEMMA 6 . 3 . ^ . - Let E be R-injective. Then the following statements are
equivalent.
1) E is indecomposable.
94
Applications of Factor Categories • . .
2) E is an essential extension of any submodule.
3) E = Eg for some irreducible right ideal K.
1
Furthermore, E , is indecomposable for a right ideal K , then
v
K
f
is irreducible.
Proof. - 1 ) ^ = ^ 2 ) It is clear from the definition.
2 ) * - * 3 ) Let x = 0 6 E. Then E pxR ^ R / ( 0 : x ) . If (0:x) • K ^ Kg ,
r
r
R/(0;x) 2 L / ( 0 : x ) « K / ( 0 : x ) . By 2) we have K. or K = (0:x) . Hence,
r
i
r
^
r
\
c.
r
(0:x) is irreducible. This proof shows the last part.
o
3)-^l) Let E^ = E,©E and p.: E
K
\ D
1
0
Ker(p |R/K). Then K
i
0
E. the projections. Put K- =
1
1
= K^n Kg. We may assume K = K-j from 3 ) . Then Ker p ^ O
since EpR/K. Hence, Eg = 0 .
THEOREM 6 . 3 . 5 [ 1 7 , 2 9 , 3 9 3 . - Let A be as above. Then A is completely
reducible if and only if for every right ideal K, K always has a
decomposition as follows : K = K ^ Kg and K
1
is irreducible and
R ^Kg i K.
Proof. - If E
K
is completely reducible, E^ = E^ % Eg by ( 6 . 3 . 1 ) and
: 6 . 3 . 2 ) , where E^ is indecomposable. Then we have K = K ^ K g and
contains an isomorphic image of R/K^ from the proof of 3) ^
1 ) of
(6.3.M.
Hence, K^ is irreducible from ( 6 . 3 . 1 * ) . Conversely, if K = K^^ Kg and
Kgf K, then we have a natural exact sequence : 0 - ^ R / K — » R/K^R/Kg and
• (Kg/K) c R/K^ Hence, E(R/K) 2E(R/K ) since E(R/K ) is indecomposable.
1
We knew already from the proof of ( 6 . 2 . 1 ) that every injective module
E contains some E^. Therefore, E contains a minimal object in A and hence
95
Applications of Factor Categories • . .
A is completely reducible, since A is a spectral, Grothendieck category.
COROLLARY 6 . 3 . 6 . - We have the following equivalent statements
1) R is a right noetherian ring.
2) Every infective modules are a direct sum of c.inde.modules.
3) Any directsums of infective modules are also infective,
29, 32]).
([3,
Proof. 1)^2)
1 ) ^ 3 ) See [ 3 ] or [ 8 ] .
Since R is right noetherian, the condition of ( 6 . 3 * 5 ) is
satisfied and so A is completely reducible. Hence, for any injective
module E, E = E(2 $ Q ) by Remark 1 and ( 6 . 3 . 2 ) , where Q *s are indecomposable and injective. Since I $ Q^ is injective, E = 1 $ Q^.
2) * * 3 ) Let {E^}^ be a set of indecomposable injective modules. We put
E = E ( U E ) . Then we have E = X 9 Qg by 2), where Qg's are indecompoa
^
J
sable and E = £ $ Q = Z $ E
. Hence, | jl = | I\ and E
R
_
J
_
is isomorphic to
—
some Qg and vice versa, since E^ and Qg are minimal in A. Therefore
I
I
9 E ^ 21 9 Q
0
is injective.
J
Remark 5 . The completely reducibility of A does not guarantee that
R is a right noetherian ( ( 6 . 3 . 3 ) ) .
reducible in general
Furthermore, A is not completely
(see fl7j).
6.1*. GOLD IE DIMENSION.
A. Goldie [ 1 5 ] defined a dimension of modules as a generalization
of noetherian modules. J. Fort [1Q] and Y. Miyashita [ 3 1 ] generalized it
96
Applications of Factor Categories . . .
independently to an infinite case. We shall reproduce them as an applica­
tion of
$.2.1).
DEFINITION.-Let M he an R-module. If M is always essential extension of
any non-zero sub-modules, M is called
uniform . Let N be an R-module.
We consider the set S of sub-modules T of N such that T = £ $ K
, where
a
K 's are uniform. Put dim N = max((ii ) if it exists (we shall show in
I
(6.U.3) that dim N exists for any N ) .
a
THEOREM 6.U.1 [ 1 0 , IT, 3 1 } • - Let E be ^-injectives. Then dim E exists
and we have a decomposition E = E • E^ such that dim E - dim E-j,
1
= 0 and E^ is a minimal injective submodule of E among
dim
1
injective submodules E of E with decompositions as above. Further­
more this decomposition is unique up to isomorphism.
Proof. - We take the factor category K in § 6 . 3 . Then dim E = 0 if and
only if the socle S(E) of E in A is zero. We assume S(E) 4 0 and S(E) =
2" § E = E(Z $ E ) , where E^'s are indecomposable injectives. Then
a
a
= 0 . Let N = I • N be a submodule in E,
J
where N ^ s are uniform. Then E(N) = E( Z. • E(N^)) and E(N ) is minimal
^
J
in A. Hence, E(N) C s(5) and so |jU |l|. Therefore, dim E * ill Let E
E = E(2 © E ) e E and dim E
a
d
d
a
1
be an injective submodule of E such that E = E'CE^', dim E
dim E
1
= 0 . Then E contains S(E) = £ • E . Hence, E
I
Ot
f
f
- dim E and
is a minimal one
1
1
among injectives with such a decomposition. Let E = E^GE^E^ 'QE^ such
that dim E
= dim E^ and dim E^ = dim E^ = 0 and E , E ' are minimal in
1
1
such decompositions. Then E = E
1
:
1
1
1
= S(E) and hence, E ^ E ^ . Since J is
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Applications of Factor Categories . . •
the radical, E ^ E ^
LEMMA
6.U.2.
-
by Remark 3 in § 1 . 1 .
and E ^ E ^ in
Let M = f 9 M
^
in L and N a submodule of M. Put
"r\
Ot
=M
Ot
and only if M p N /br all ot£ I,
f
and H = Z ® N . Then M ? N
^
N
Ot
ot
Ot
ot
1
(further, M p N ) •
Proof. - Suppose M ? K
ot
for all a . Let m i oQ M ; m = Em
oCM •
i
i
^i
r+ ^ m r and
ot
Trom the assumption, there exists r€R such that mr = m
a
i
i*2
i
!
m^ r 4
• Repeating this, we obtain mHON' ± 0 . Hence, M ;pN . The
converse is clear.
PROPOSITION 6.U.3. - Let N be an ^-module. Then dim N exists and N is an
essential extension of a submodule N^GNg such that dim N^=dim N = I I I
and dim N = 0 and N is an essential extension of E * T
2
T^'s
1
E
1
and N S N ^ O T g
1
E
a s
A
1
(6.U. 1 ) . Put 1SL = N /OE^.
i n
1
2
by ( 6 , ) + . 2 ) . Hence, dim E = dim N
1
2
!
E
> where
are uniform.
Proof. - Put E = E(N). Then E = ^
Then E^2
a
= 0 and
!
= E(N„ ). Let E„ = E(£$E ), where E s are indecomposable. Put
1
1
a
ot
f
E O N / = N and N = £ 9 N
ot
1
a
l ^ a
(6.U.2).
Suppose N 7 T
!
f
. Then N s are uniform and N. :> N by
1
ot
= Z 9T
a
J _
1
f
, where T s are uniform. Then
a
_
E(T ) = E ( I 9 E(T )) = £6E(T ) C E . Hence, [jU H I and dim N = dim E = \Jt.
%J
!
a
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Applications of Factor Categories . • •
COROLLARY 6.k.h. £ 9 ] . - Let
Q = Z • E
I
a
a
be
set of injective modules and
!
. Let P be a submodule of Q such that P * £ $ P ; P s
J
R
g
are indecomposable injectives. Then IJl^Ill.
6 . 5 . THE PROPERTY III IN INJECTIVE MODULES.
In this section we shall study the property III in a case where
every c.inde. modules are injective, which is called Matlis'problem £29].
We do not know a complete answer for this problem and we shall give here
some affirmative answers given by ["25] ^d [ U o ] .
a
From the proof of ( 6 . 2 . 2 ) we have
LEMMA 6 . 5 . 1 . - Let {N }j be a set of indecomposable injectives. If
Z(N^) = 0 for some a , every non-zero element in j^N^ .N "1 is
isomorphic. Especially, if Z(N ) = 0 for all a^f*, {N )j is a
a
a
T-nilpotent system with respect to J!
THEOREM 6 . 5 . 2 [ 2 1 , 2 5 , ho] . - Let {N }j be a set of indecomposable
injectives and N = ¿ 1 ^ . Suppose N = M ^ M ^ and Z(M^) = 0.
Then M^
t s
<2 dizectsum of c.inde. injectives for i - 1 , 2 .
Proof. - M^ contains a dense submodule T^ by ( 3 . 2 . 7 ) . Let T ^
^T^T^s
are c.inde.. Since Z(M^ = 0 , Z(T^ = 0. Hence, (T )j is a T-nilpotent
system by ( 6 . 5 - 1 ) . Therefore, we have the theorem from ( 3 . 2 . 2 ) and
(I*. 1 . 3 ) .
99
Applications of Factor Categories . . .
THEOREM 6 . 5 . 3 . - Let { }j ^
E
e
a
a
and E = X ®
E
a
s e t
°f indecomposable injective modules
• Then the followings are equivalent.
1
1) {E^}^ is a locally semi-T-nilpotent system with respect to J .
2) Every module in £ which is an extension of E contains E as a
direct summand.
3) There are no proper and essential extension of E which core in
C.
4) For each monomojuphism g in S
£
= End(E), Im g is a direct
summand of E,
where £ is the category of all c.inde.modules.
1 ) It is proved by ( U , 1 . 5 ) .
Proof. - U)
1) ^
h) Let g be a monomorphism in S . Then Im g = Z ©g(E ) and E
g(E ).
Since g(E ) are injective, Im g is a locally direct summand of E. Hence,
a
Im g is a direct summand of E by 1 ) and ( 3 . 2 . 5 ) »
1)
2 ) It is clear from the above proof.
2) ^
3 ) It is also clear.
3) ^
1 ) Suppose {E^}^ is not a locally semi-T-nilpotent. Then there exist
a subset
{E^ K of (E_] and a set of non-isomorphisms f. : E -*> E
T
1
i
a
i
a
1
such that for some element x in E
f f „ ... f.(x)^ 0 for all n. We
cx^ n n-i
I
note Ker f^ 4 0 , since E^ are injective and indecomposable. Put
i
00
E. = j x.+f. (x. ) I x.«! \ £ F
1
( , 1 1 1 lot.
1
j
1
1
§ E
a.
1
i +
CO
< • E. Put E = H « E
9 E
• ot.
o
1=1
100
1
%
Applications of Factor Categories • . .
E
H (Z • E ^ 2 Ker f \ # 0 . Hence, Z H ' .
F
A
e E C E by (6.U.2). It is
q
F
clear x ^ ( Z ® E . 8 E ). Let E * be an injective hull of E . Since
!
(£ $ E . $ E ) «
J
o —^
t
E , we can extend this isomorphim t to a monomorphism
<J) of E* .Therefore, <j>(Z « E» . ® E ) = E £ <(>(E) = £$$(E )<; C. which is
j
o
T
I
a contradiction.
COROLLARY 6 . 5 . ^ . - Let { E ^ and E be as above. Furthermore> we assume that
each E is noetherian. Then all statements in
a
Proof. - Let (E^}*J
(6.5.3) are true.
be a set of injective and indecomposable modules and
f. : E. —* E. non-isomorphisms. Then Ker f. 4 0 , Im f,n Ker f
l
l
1+1
*
l
1
2
i
f
# 0 . Since E
i
0
2
is uniform. Hence, Ker f ^ K e r fgf » if
1
0 if
0- Therefore,
{E^}^ is a T-nilpotent system form the assumption.
COROLLARY 6 . 5 - 5 . Let M be a module in C
and L a submodule of M. Suppose
L is a direct sum of infective modules and Z(L) = 0 . Then L is a
direct summand of M
Proof. -
(cf. [ 9 , 2 1 , 2 5 ] ).
Since every injective module in M is in C by ( U . 1 . 5 ) , the
corollary is clear from ( 6 . 5 - 3 ) .
Remark 6 . Let {E^} be as in ( 6 . 5 . 3 ) . I n general {E^}^ is not semi-T-nil­
potent. Hence E = [ I E
I
is not quasi-injective. Furthermore, even if
a
E ^ are noetherian, E is not injective. If E is (quasi-)injective or Z(E)=0
{ E } is semi-T-nilpotent. However, the converse is not true (see [**2]).
A
101
Applications of Factor Categories . . .
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Manuscrit remis an mai 1974.
Manabu HARADA
DEPARTMENT OF MATHEMATICS
OSAKA CITY UNIVERSITY
OSAKA
104
Japan.